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Transformations on Willmore surfaces

Katrin Leschke

Habilitationsschrift zur Erlangung der Venia Legendi, eingereicht an dermathematisch–naturwissenschaftlichen Fakultat der Universitat Augsburg

17. Mai 2006

Contents

0 Introduction 5

1 Quaternionic Holomorphic Geometry 13

1.1 Holomorphic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.1 Conformal maps into the 4–sphere . . . . . . . . . . . . . . . . . . 14

1.1.2 Holomorphic curves in HPn . . . . . . . . . . . . . . . . . . . . . . 24

1.1.3 Frenet curves in HPn . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.2 Holomorphic line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.2.1 Holomorphic vector bundles . . . . . . . . . . . . . . . . . . . . . . 35

1.2.2 The Kodaira correspondence . . . . . . . . . . . . . . . . . . . . . 38

1.2.3 The Riemann–Roch Theorem . . . . . . . . . . . . . . . . . . . . . 41

1.2.4 The Plucker formula . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2 Transformations on conformal maps 47

2.1 The Darboux transformation . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.1.1 The classical Darboux transformation on isothermic surfaces . . . 48

2.1.2 The spectral curve of constant mean curvature tori . . . . . . . . . 52

2.1.3 The Darboux transformation on conformal maps into the 4–sphere 55

2.1.4 The Darboux transformation on constant mean curvature surfaces 58

2.1.5 The spectral curve of a conformal torus . . . . . . . . . . . . . . . 62

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4 CONTENTS

2.1.6 The Darboux transformation on holomorphic curves . . . . . . . . 63

2.2 The Backlund transformation . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.2.1 The Backlund transformation on holomorphic curves . . . . . . . . 65

2.2.2 Construction of Backlund transforms from Abelian integrals . . . . 66

2.2.3 Envelopes and Osculates . . . . . . . . . . . . . . . . . . . . . . . . 68

2.2.4 The (n+1)–step Backlund transformation . . . . . . . . . . . . . . 72

2.2.5 The backward Backlund transformation . . . . . . . . . . . . . . . 77

3 Applications to Willmore curves 81

3.1 Willmore curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.2 The Darboux transformation on Willmore surfaces in S4 . . . . . . . . . . 87

3.3 The Backlund transformation on Willmore curves . . . . . . . . . . . . . . 88

3.3.1 Sequences of Willmore curves . . . . . . . . . . . . . . . . . . . . . 89

3.3.2 Finite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3.3 Willmore spheres and Willmore tori . . . . . . . . . . . . . . . . . 94

Chapter 0

Introduction

In surface theory one frequently considers surface classes which come by variational princi-ples: one tries to find all surfaces which are critical points of a certain “energy” associatedwith the surface. The most famous examples are the minimal surfaces which are criticalpoints of the area functional. Given a boundary curve, one tries to find the surface withthe least energy having this boundary. Further examples of surface classes arising fromvariational principles are constant mean curvature surfaces which are the critical pointsof the area functional under the constraint to preserve the volume, and the Willmoresurfaces, the critical points of the bending energy W(f) =

∫H2 where H is the mean

curvature of f : M → R3.

Over the past 25 years significant progress has been made in the classification and con-struction of these surfaces, e.g. [Cal67], [Bry82], [Bry84], [Hit90], [PS89], and [Bob94].The theory of each of these surface classes is closely linked to the theory of harmonic mapsinto some associated symmetric space: in the case of a minimal surface, the harmonic mapis given by a conformal parametrization, for constant mean curvature surfaces it is givenby the Gauss map, and for Willmore surfaces by the conformal Gauss map. The harmonicmap equations for the various cases turn out to be completely integrable partial differentialequations.

In the simplest case these equations lead to solutions given in terms of holomorphic func-tions, for example the classical Weierstraß representation of minimal surfaces. In the caseof tori, solutions to the harmonic map equations can be obtained from theta functionson an auxiliary Riemann surface, the so–called spectral curve [PS89], [Hit90], [Bob91],[McI01]. For some surface classes the spectral curves have finite genus (e.g. CMC toriand Willmore tori), which allows for explicit parametrizations in terms of theta functionsof all the tori in this class [Bob91], [Sch02], [FPPS92].

Despite of these results, a number of basic questions remain unanswered. For instance,what are the minimal values of the variational problems, or on which tori are they attained.For higher genus the situation is even more unsatisfactory: there are only few explicit

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6 CHAPTER 0. INTRODUCTION

examples of higher genus constant mean curvature or Willmore surfaces, and little isknown about an appropriate generalization of the “spectral curve”.

The analytic difficulties come from the fact that the fundamental equations of a surface in3-space, the Gauss-Codazzi equations, are a nonlinear, third order system. Moreover, theGauss-Codazzi equations for various classes of surfaces give rise to qualitatively differentsystems of differential equations, each demanding its own theory.

In comparison, the theory of holomorphic curves into complex projective space is veryrich: explicit examples for every genus are studied, global properties and methods applied.One of the reasons for the difference to surface theory is that the fundamental equationof algebraic curve theory is the linear, first order Cauchy-Riemann equation, and muchmore can be said about solutions. Moreover, the Kodaira correspondence gives the linkbetween holomorphic curves and holomorphic line bundles, and the theory of algebraiccurves can be formulated in the language of holomorphic line bundles. Cornerstones forthe theory of algebraic curves are the Riemann-Roch Theorem, the Clifford estimate, thePlucker relations and the Abel map.

Over the past years, a completely new theory, the so–called Quaternionic HolomorphicGeometry, [PP98], [BFL+02], [FLPP01], has been built to combine the theory of algebraiccurves with the theory of conformal maps of a Riemann surface into 3– or 4–space. InQuaternionic Holomorphic Geometry, the conformal geometry of the 4–sphere is modeledby the projective line HP1 on which the group of orientation preserving Mobius transfor-mations acts by Gl(2,H). The holomorphic maps in this theory are exactly the conformalmaps into the 4–sphere. The appeal of this model is that the geometric background givesinsights in how to build a quaternionic extension of Complex Analysis, and, conversely,well–known results for complex holomorphic curves in CPn can be translated to the quater-nionic setting and give, when applied to surfaces in 3– and 4–space, new results in surfacetheory.

In Chapter 1 we give a fairly coherent summary of Quaternionic Holomorphic Geometry:on one hand it will help the reader to get a flavor of the ideas and motivations of thetheory, on the other hand it will setup the notations and tools needed for the results ontransformations of conformal maps.

Conformal maps f : M → R3 from a Riemann surface M with complex structure J into3–space satisfy a Cauchy–Riemann–type equation df(JX) = Ndf(X) with varying “i”.Here, N : M → S2 is the normal to the surface and the multiplication is the multiplicationin the quaternions where we identify R3 with the imaginary quaternions Im H. If f takesvalues in a 2–plane its unit normal is a constant map, say N = i, in which case we recoverthe usual Cauchy–Riemann equation for a holomorphic map f : M → C.

Using this observation as a starting point, the quaternionic holomorphic theory can bederived analogous to the complex case [PP98],[BFL+02], [FLPP01]: the holomorphic func-tions in the quaternionic setting are the conformal maps into the 4–sphere. A key obser-vation is that a map f : M → S4 = HP1 from a Riemann surface M into the 4–sphere

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is the same as the quaternionic line bundle L over M whose fiber at p ∈ M is givenby Lp = f(p). We explain how the quaternionic setup can be used to study conformalgeometry of surfaces in 3– and 4–space. In particular, the basic notions for conformalimmersions from a Riemann surface into the 4–sphere and for surfaces in Euclidean spaceare expressed in terms of quaternionic calculus, e.g., the mean curvature sphere, the Hopffields, the Willmore functional, the mean curvature vector, and the Gauss and normalcurvature.

The conformality of f gives rise to a complex structure J and a holomorphic structure onthe dual line bundle L−1. Here a quaternionic holomorphic structure on a quaternionicvector bundle V with complex structure J is a quaternionic linear map D : Γ(V )→ Γ(KV )which satisfies the Leibnitz rule D(ψλ) = (Dψ)λ+(ψdλ)′′ for λ : M → H where ω′′ denotesthe (0, 1)-part of a 1–form ω.

The complex Kodaira correspondence translates to the quaternionic setup: a quaternionicholomorphic curve f : M → Gk(Hn) into the k–plane Grassmannian corresponds, up toMobius equivalence, to a basepoint free n–dimensional subspace H ⊂ H0(V ) of the spaceof holomorphic sections of a k–dimensional quaternionic holomorphic vector bundle V[FLPP01]. In particular, a holomorphic curve [f1 : f2 : . . . : fn+1] : M → HPn from aRiemann surface into HPn gives rise to a family of conformal maps: the coordinate mapsfi : M → H are conformal.

Moreover, the Riemann–Roch theorem holds verbatim for quaternionic holomorphic vec-tor bundles [PP98], [FLPP01]. However, the quaternionic Plucker formula involves a newquaternionic invariant: the Willmore energy W(D) of the holomorphic structure D. Aquaternionic holomorphic structure decomposes D = ∂ + Q in J commuting and anti-commuting parts, and the Willmore energy is given by W(D) = 2

∫M < Q ∧ ∗Q >. In

particular, W(D) measures the deviation from the complex case: for Q ≡ 0 we recoverthe theory of complex vector bundles. If the holomorphic structure is induced via theKodaira embedding of a conformal immersion f : M → S4 of a compact Riemann surfaceinto the 4–sphere then the Willmore energy of D is exactly the classical Willmore func-tional W(f) =

∫|H|2 −K −K⊥ where H is the mean curvature vector and K and K⊥

are the Gauss and normal curvature of f .

As mentioned before, applying results of this theory to surfaces in 3– and 4–space givessubstantial insight into classical problems of surface geometry: the quaternionic Pluckerformula provides lower bounds for the energy of harmonic tori in the 2–sphere and the areaof constant mean curvature tori in 3–space in terms of their spectral genus. In particular,for a constant mean curvature torus of spectral genus > 6 the Willmore and Lawsonconjectures are both satisfied [FLPP01].

Transformations which preserve special surface classes in 3– and 4–space play an importantrole in surface geometry. One of the motivations for the study of these transformationscomes from the fact that they allow to construct more complicated surfaces from givensimple surfaces. Historical examples include the Backlund transformation on surfaces


of constant Gaussian curvature [Bia80] and the Darboux transformation on isothermicsurfaces [Dar99].

In Chapter 2 we discuss a generalized Darboux and Backlund transformation on conformalmaps f : M → S4 of a Riemann surface into the 4–sphere. To define the Darbouxtransformation we use a geometric construction [BLPP]. A sphere congruence S envelopesa conformal immersion f : M → S4 if for all p ∈ M the sphere S(p) passes through f(p)and the oriented tangent spaces of f and S(p) coincide. It is a classical result [Dar99] thatif f : M → R3 allows a sphere congruence enveloping f and a second surface f ] : M → R3

then both f and f ] are isothermic, and f ] is a Darboux transform of f . To generalizethis transformation it is necessary to refine the enveloping property: we say that thesphere congruence S left-envelopes f if S(p) goes through f at p and the oriented tangentspaces are left–parallel, that is if their associated oriented great circles on S3 correspondvia left translation in the group S3. Given a conformal map f : M → S4 a conformalmap f ] is called Darboux transform of f if there is a sphere congruence enveloping f andleft–enveloping f ].

The spectral curve of a conformal torus f : T 2 → S4 with trivial normal bundle can bedefined [BLPP], loosely speaking, as the set of all closed Darboux transforms of f . Foreach point p ∈ T 2, the images f ](p) of the Darboux transforms f ] of f canonically embedthe spectral curve into S4 as a twistor projection of a holomorphic curve F (p, .) : Σ→ CP3.It can be shown that Σ is a Riemann surface of possibly infinite genus.

In the case of a constant mean curvature torus f : T 2 → R3 this definition of the spectralcurve coincides [CLP] with the “classical” one given by the eigenvalues of the holonomy ofa family of flat connections [Hit90]. We show in Chapter 2 that the Darboux transformscorresponding to points on the spectral curve are isothermic even though the generalDarboux transformation only coincides for very special points on the spectral curve withthe classical Darboux transformation of a constant mean curvature torus in R3.

To define [LP03] a Backlund transformation on conformal maps f : M → S4, recall theenveloping and osculating construction for conformal maps into the 4–sphere, or moregeneral, for holomorphic curves into HPn. The first osculate of a holomorphic curvef : M → HPn is obtained by intersecting the tangent of f with a fixed hyperplane.Conversely, an envelope f of f is given by integrating prescribed tangents so that f is thefirst osculate of f . In [LP05] we extend the Backlund transformation to holomorphic curvesin HPn, and show that the Backlund transform is given by an envelope of f . We use thisgeometric picture to show in Chapter 2 a Bianchi permutability theorem. In particular,Backlund transforms of Frenet curves are Frenet, and the (n+1)–step Backlund transformf of a holomorphic curve f : M → HPn can be computed solely by differentiation andalgebraic operations so that the (n+ 1)–step Backlund f is globally defined.

The study of Willmore surfaces, the critical points of the bending energy∫H2 of a sur-

face, goes at least back to Blaschke’s school in the 1920’s [Bla29]. About 40 years laterWillmore [Wil68] reintroduced the problem and focused on minimizers for the bending

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energy, nowadays called the Willmore energy, over compact surfaces of fixed genus. Heshowed that the round sphere is the minimum among genus 0 surfaces and formulated theconjecture that the minimum over tori is given by the Clifford torus with Willmore energy2π2. In the 1980’s Bryant [Bry84] classified all Willmore spheres in 3-space as invertedminimal spheres with planar ends in R3. Subsequently Ejiri [Eji88] and recently Montiel[Mon00] proved an analogous result for Willmore spheres in 4–space — in addition toinverted minimal spheres in R4 we also have twistor projections to S4 of rational curves inCP3. The case of Willmore tori is more involved: To a Willmore torus f : T 2 → S4 withtrivial normal bundle one can associate its spectral curve, namely the Riemann surfacedefined by possible monodromies of the associated S1–family of flat connections [FPPS92].The spectral curve of a Willmore torus has finite genus [Sch02], and the Willmore torusis then parametrized by theta functions on the spectral curve, [FPPS92], [Sch02]. In fact,the recent preprint [Sch02] by Schmidt seems to go some way toward proving the Willmoreconjecture.

In Chapter 3 we study Willmore curves in HPn: A holomorphic curve f : M → HPn froma compact Riemann surface into HPn is Willmore [LP03] if f is a critical point of theWillmore energy under variations of f which have at least n+ 1 holomorphic sections inthe associated holomorphic line bundle of f .

An important aspect of the theory of Willmore surfaces is its connection to harmonicmaps: the conformal Gauß map or mean curvature sphere congruence S of a Willmoresurface is harmonic. Similar to the ∂ and ∂ sequence of harmonic maps h : M → CPnthe (0, 1) and (1, 0)–part Q and A of the derivative of the conformal Gauss field give new(possibly branched) conformal immersions f , f : M → S4 provided A 6≡ 0 and Q 6≡ 0. Weshow that the conformal Gauss maps of f and f extend smoothly into the branch points,and both surfaces are Willmore.

A degree computation shows that all mean curvature spheres of a Willmore sphere f :S2 → S4 go through a constant point q ∈ f(S2) on the surface if A,Q 6≡ 0. Thestereographic projection across q gives a minimal surface in R4. In the case when A ≡ 0(or Q ≡ 0) then f (or its dual curve) is a twistor projection of a holomorphic curve in CP3

[FLPP01], and we recover Montiel’s result [BFL+02]. In [Les] this result is generalized toWillmore spheres in HPn: a Willmore sphere in HPn has integer Willmore energy and isgiven by complex holomorphic data [Les].

The case of Willmore tori is more involved: there are examples of Willmore tori in the4–sphere constructed by integrable system methods which are neither inverted minimalsurfaces nor twistor projections of elliptic curves [Pin85], [FP90], [BB93]. However, if theWillmore torus f has non–trivial normal bundle then we showed in [LPP05] that f comesfrom the twistor projection of a holomorphic curve in CP3 or from a minimal surfacein R4. The method of proof in [LPP05] is to examine the possible monodromies of theassociated S1–family of flat connections. In Chapter 3 we present a different approach byusing sequences of Willmore surfaces [LP]. This approach can be extended to the case ofWillmore tori in HPn with degL 6= 0, and we show that such a Willmore torus has integer


Willmore energy.

Acknowledgements: First of all, I would like to thank the members of my FachmentoratJ. Dorfmeister, K. Grosse-Brauckmann, E. Heintze, and J. Ritter for advising me in thewhole process of the Habilitation.

Moreover, I’m deeply grateful to my collaborators over the last couple of years:

D. Ferus not only supervised my Ph. D. thesis but also led my way to QuatenionicHolomorphic Geometry. From the beginning of my studies as a student up to today,he was always an excellent teacher of mathematics, a huge influence on my work, andsomeone I could turn to for advice. Without his constant encouragement and support,this Habilitation thesis could never had happened.

F. Burstall legendary appearances at the H–Seminar in Berlin sharpened my view beyondthe quaternionic theory. His deep insight into conformal geometry and his willingness toshare his knowledge are exemplary.

The discussions with E. Carberry on the spectral curve of a conformal torus, and herpatient explanations on algebraic curve theory clarified my picture of the spectral curve.

U. Pinkall taught me early on in my studies to trust my mathematical instinct and to seeka deep understanding of a problem instead of being satisfied with a formal argument —searching for the deep reasons reveals the beauty of mathematics and leads to interestingresults. His intuition is at the bottom of all our joint work, and influences my researchup to now. Thanks to him and F. Pedit for letting me join the exciting explorations ofQuaternionic Holomorphic Geometry.

F. Pedit’s impact on my work is tremendous — it is impossible to list everything I owe tohim. Besides his huge knowledge, his keen sense for mathematics and his inspirited ideas,I admire his capacity to boil down an argument to its core: I profited from this abilitynot only through his C–seminar and H–seminar lectures but also in many discussions onthe quaternionic theory later on. Working with him certainly changed the way I attack amathematical problem as well as how I present and teach mathematics to the better. Hisenergy and enthusiasm for mathematics made our discussions very exciting, and turnedour work in Amherst into “the good life”.

Moreover, I would like to thank my colleagues at the Technische Universitat Berlin, theUniversity of Massachusetts, and Universitat Augsburg for many fruitful discussions andinterest in my work, in particular, the participants of the H–Seminar in Berlin, the GANG-Seminar in Amherst, and the Oberseminar ”Integrable Systeme” in Augsburg. At mytime in Berlin, I taught a course explaining the first steps of Quaternionic HolomorphicGeometry and early results presented in this thesis: I would like to thank the students

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of this class as well as the participants of the H–Seminar–“Nachsitzungen”, in particular,C. von Renesse, U. Heller, P. Peters, and P. McDonough, for their interest and for all thefruitful and amiable discussions on the topic. Special thanks goes to U. Heller: his Ph.D. project involved to find examples of higher genus Willmore surfaces in 4-space usingcomplex holomorphic curve theory in CPn and methods from Quaternionic HolomorphicGeometry, and we spent hours in front of a computer to find and discuss pictures ofWillmore spheres and Willmore tori. Some of the pictures here are taken from his Ph. D.thesis, and none of the pictures in this work would have been possible without his and M.Schmies’ help.

Finally, I would like to thank my parents, my family, and my friends for their constantsupport and love.

Chapter 1

Introduction to QuaternionicHolomorphic Geometry

In this chapter, we review the main techniques and results developed in QuaternionicHolomorphic Geometry, [PP98], [BFL+02], [FLPP01]. The objective is to use a “quater-nionified complex analysis” to obtain results in surface theory. We generalize the theoryof algebraic curves h : M → CPn of a Riemann surface M into complex projective spaceto a theory of holomorphic curves f : M → HPn into quaternionic projective space. Thisis geometrically motivated by the observation that the conformality condition for a mapf : M → S4 is, when expressed in terms of quaternions, a Cauchy-Riemann equationwith varying “i” on the target space. Here, we use the fact that the conformal geometryof the 4–sphere can be modeled by the quaternionic projective space HP1 = S4 wherethe group of orientation preserving Mobius transformations is given by GL(2,H). Var-ious results of algebraic curve theory can be extended to the quaternionic setting. TheKodaira correspondence links holomorphic curves in HPn with quaternionic holomorphicline bundles. Analytically, the Cauchy–Riemann operator ∂ on a complex vector bundleis replaced in the quaternionic theory by an elliptic operator ∂ + Q on a quaternionicvector bundle L with complex structure J , where Q is a (0, 1)–form with values in theJ–complex antilinear endomorphisms of L. The L2–norm of Q provides an invariant ofthe quaternionic theory, the Willmore energy of the quaternionic holomorphic structure∂+Q. The properties of quaternionic holomorphic structures are similar to the propertiesof a ∂ operator in the complex case: the vanishing orders of holomorphic sections arewell–defined, the Riemann–Roch theorem holds verbatim, and the Plucker formula givesestimates on the Willmore energy.

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14 CHAPTER 1. QUATERNIONIC HOLOMORPHIC GEOMETRY

1.1 Holomorphic curves in HPn

We use the example of a conformal map f : M → R3 to motivate the definition of aholomorphic curve f : M → HP1, and more general, of a holomorphic curve in HPn.The conformal Gauss map of a conformal map f : M → S4 will turn out to be animportant ingredient to built a theory of quaternionic holomorphic curves and quaternionicholomorphic vector bundles. The conformal Gauss map gives a complex structures S onthe trivial H2–bundle V = H2. Generalizing the conformal Gauss map, we obtain theso–called canonical complex structure of a holomorphic curve f : M → HPn. In general,the conformal Gauss map of a conformal map f : M → S4 does not extend into the branchpoints of f . A holomorphic curve for which the mean curvature sphere congruence, ormore generally the canonical complex structure, extends to M is called a Frenet curve.The Willmore energy of a Frenet curve is the analog of the Willmore energy W (f) =

∫H2

of a conformal immersion f : M → R3 where H is the mean curvature of f .

1.1.1 Conformal maps into the 4–sphere

We use the example of a conformal immersion f : M → R3 of a Riemann surface M into3–space to motivate how to define holomorphic curves in S4 = HP1 or, more generally,in HPn. The Riemann surface M comes with a complex structure JTM which gives the90 degree rotation in the tangent space TM . In particular, an immersion f : M → R3 isconformal if and only if for all X ∈ TM :

df(JTMX) ⊥ df(X) and |df(JTMX)| = |df(X)| . (1.1)

Choosing the sign of the unit normal vector N appropriately, (1.1) reads as

df(JTMX) = N × df(X) . (1.2)

Figure 1.1: Conformal maps into 3–space

We rewrite (1.2) in terms of quaternions: The Euclidean 3-space can be identified withthe imaginary quaternions

im H = SpanRi, j, k = R3 ⊂ R4 = SpanR1, i, j, k

where i2 = j2 = k2 = −1. The conjugation in the quaternions is defined by

x = x0 − x1

for x = x0 + x1 with x0 ∈ R, x1 ∈ im H, so that

x ∈ im H if and only if x = −x .

1.1. HOLOMORPHIC CURVES 15

The inner product on R4 is given in terms of quaternions as

< x, y >= Re (xy)

so that the cross product for orthogonal vectors in R3 becomes the quaternionic multipli-cation. In particular, (1.2) reads as

df(JTMX) = Ndf(X) .

Denoting by ∗ : Ω1(TM)→ Ω1(TM) the negative Hodge star, i.e.,

∗df(X) = df(JTMX), for all X ∈ TM ,

we see that an immersion f : M → R3 is conformal if and only if

∗ df = Ndf (1.3)

with N : M → S2. Note that

S2 = N ∈ im H | N2 = −1

since |N |2 = NN = −N2. If f : M → R3 is a conformal immersion, then

− ∗ df = ∗df = Ndf = df N = dfN

shows that∗ df = Ndf = −dfN . (1.4)

Furthermore, we observe that (1.3) is a Cauchy–Riemann equation with varying “i”: iff : M → C = Spanj, k ⊂ im H, then f has constant normal N = ±i and f is conformalif and only if ∗df = ±idf , i.e., f is holomorphic or antiholomorphic. An antiholomorphicmap can be interpreted as a holomorphic map if we equip C with the complex structure−i instead of i. This leads to the following definition:

Definition 1.1. A map f : M → R3 is called holomorphic if there exists N : M → S2

with ∗df = −dfN .

Note that we dropped the assumption that f is immersed. However, a holomorphic mapis conformal. We will see below that if df does not vanish identically, the holomorphicityof f : M → R3 implies that df has only isolated zeros. In particular, f is branchedconformal immersion. To generalize our notion of holomorphicity to maps f : M → R4

into the 4–space, recall the Fundamental Lemma:

Lemma 1.2 (Fundamental Lemma, e.g. [BFL+02, Lemma 2]). Every 2–dimensionalsubspace E ⊂ R4 = H is given by a pair (N,R) ∈ S2 × S2 by

E = x ∈ H | Nx+ xR = 0

N and R are called the left normal and right normal of E. The pair (N,R) is unique upto sign. The orthogonal complement E⊥ of E is given by

E⊥ = x ∈ H | Nx− xR = 0 .


Note that the left and right normal vector of E are in general not orthogonal to E.However, since N,R ∈ S2, we have Nx, xR ∈ E and Nx ⊥ x and xR ⊥ x for all x ∈ E aswell as |Nx| = |xR| = |x|.

Applying the Fundamental Lemma in the case of an immersion f : M → R4 = H, we seethat the tangent space and normal space of f are given at each point p ∈ M by a pair(N,R) ∈ S2 × S2 via

dpf(TpM) = x ∈ R4 | Nx+ xR = 0 (1.5)

and⊥p M = x ∈ R4 | Nx− xR = 0 . (1.6)

Corollary 1.3. Let f : M → R4 = H be an immersion. Then f is conformal if and onlyif there exist N,R : M → S2 with

∗ df = Ndf = −dfR . (1.7)

If f : M → R3 = im H, then R = N is a unit normal vector field to f .

Remark 1.4. Note that in the case of an immersion f : M → H the two conditions

(1) there exists N : M → S2 with ∗df = Ndf

(2) there exists R : M → S2 with ∗df = −dfR

are equivalent. In particular, in Corollary 1.3 we could replace (1.7) by either one of theconditions (1) or (2).

Since we consider conformal maps up to Mobius equivalence, we extend our interest toconformal maps f : M → S4 of a Riemann surface into the 4–sphere. We model theMobius geometry of S4 by the projective geometry of the (quaternionic) projective lineHP1: A map f : M → S4 is given by the line subbundle L ⊂ V of the trivial H2–bundle V over M , where the fiber Lp over p ∈ M is the projective point Lp = f(p). Inother words, L = f∗T where T is the tautological bundle over HP1. In what follows,all vector spaces are, if not mentioned otherwise, quaternionic right vector spaces, andHom(V,W ) denotes the space of quaternionic linear maps from a (quaternionic) vectorspace V to a (quaternionic) vector space W . Since the tangent space of HP1 is given byT[v]HP1 = Hom([v],H2/[v])), the derivative of f : M → HP1 is given by

δ = πd|L ∈ Ω1(Hom(L, V/L)) , (1.8)

where π : V → V/L and d is the trivial connection on V = M × H2. In particular,if f : M → H is a conformal immersion with ∗df = −dfR, then the line bundle L ofg = [f, 1] : M → HP1 is given by L = ψH where

ψ =(f1

).


Moreover, the derivative of g is

δψ = π

(df0

).

Since f is conformal we can equip the line bundle L with the complex structure J ∈Γ(End(L)), J2 = −1, defined by

Jψ = −ψR .

For ϕ ∈ Γ(L) we have ϕ = ψλ for some quaternionic valued function λ : M → H, andtherefore, away from the zeros of λ,

Jϕ = −ϕλ−1Rλ .

Moreover,

∗δψ = π

(∗df0

)= −π

(dfR

0

)= δJψ ,

shows that ∗δ = δJ for a conformal immersion g : M → H. This observation motivatesthe following definition:

Definition 1.5. A map f : M → HP1 is called a holomorphic curve if there exists acomplex structure J ∈ Γ(End(L)) with

∗δ = δJ .

Corollary 1.6. Let f : M → H be an immersion and g = [f, 1] : M → HP1. Then f isconformal if and only if g is a holomorphic curve.

After choosing a point at infinity on S, an oriented round 2–sphere S in HP1 is given inaffine coordinates by an oriented affine 2–dimensional subspace E ⊂ H. By the Funda-mental Lemma there exist (N,R) ∈ S2 × S2 and H ∈ H with

E = x ∈ H | Nx+ xR = H

with NH = HR. Thus, in affine coordinates,

S = [x, 1] | Nx+ xR = H ∪ [1, 0] .

If we consider

S =(N −H0 −R

)then S2 = −1 due to NH = HR and N2 = R2 = −1, in other words S ∈ End(H2)is a complex structure. Moreover, S[v] = [v] if and only if [v] ∈ S. Therefore, anoriented round 2–sphere S in HP1 can be identified with a complex structure S ∈ End(H2),S2 = −1, via

S = [v] ∈ HP1 | S[v] = [v], v ∈ H2 .


The line bundle L of the conformal map S is S–stable, and the conformality equation ofthe embedded round sphere S is ∗δ = δS where δ is the derivative of S. Moreover, Sinduces a complex structure on V/L and

∗δ = δS = Sδ .

A sphere congruence assigns to each point p ∈ M an oriented round 2–sphere S(p). Inother words, a sphere congruence is a complex structure S ∈ Γ(End(V )), S2 = −1, on V .We say that a sphere congruence S ∈ Γ(End(V )), S2 = −1 envelopes a conformal mapf : M → HP1, if for all points p ∈ M the sphere S(p) passes through f(p), and if theoriented tangent space of f(p) coincides with the oriented tangent space of S(p) at f(p)over immersed points p ∈M , i.e.:

SL = L and ∗ δ = Sδ = δS , (1.9)

where L = f∗T is the line bundle of f .

Figure 1.2: S envelopes f

Definition 1.7. Let f : M → HP1 be a holomorphic curve. Then S ∈ Γ(End(V )), S2 =−1, is called an adapted complex structure if

∗δ = Sδ = δS .

Adapted complex structures are not unique: if S and S are adapted, then

R = S − S ∈ Γ(R−) ,

whereR = R ∈ End(H2) | R|L = 0, imR ⊂ L , (1.10)

and, denoting by End±(V ) the S commuting and anticommuting endomorphisms,

R− = End−(V ) ∩R = R ∈ R | SR = −RS .

To single out a particular adapted complex structure, we recall the mean curvature spherecongruence (also called the conformal Gauss map) of a conformal immersion f : M → Hwhich plays the role of the Gauss map in Mobius geometry: To each point p ∈M , assignthe sphere S(p) so that S(p) goes through f(p) with coinciding oriented tangent spaces,and S(p) has the same mean curvature vectorH(p) as f at f(p). To compute the conformalGauss map of a conformal immersion f : M → S4, we first recall (1.5) that

Ndf(Y ) + df(Y )R = 0 ,


for X,Y ∈ Γ(TM) and thus

dN(X)df(Y ) +N(X · df(Y )) + (X · df(Y ))R− df(Y )dR(X) = 0 .

Using the above equation we obtain the second fundamental form II of f as

2II(X,Y ) = 2(X · df(Y ))⊥ = (X · df(Y )−NX · df(Y )R)= ∗df(Y )dR(X)− dN(X) ∗ df(Y ) .

The mean curvature vector H of f is given by H = 12 tr II so that

4|df(X)|2H = II(X,X) + II(JX, JX) = (−df ∧ dR− dN ∧ df)(X,JX) .

Since dN ∧ df = d(Ndf) = −d(dfR) = df ∧ dR and

df ∧ dR(X, JX) = df(X)(∗dR(X) +RdR(X))

we obtain 2dfH = −(∗dR+RdR) and

2Hdf = ∗dR+RdR . (1.11)

Defining H by H = HN we get

2Hdf = dR−R ∗ dR . (1.12)

Note that for a conformal immersion f : M → R3 the mean curvature of f is given byH = −HN . A similar computation with N instead of R gives

−2dfH = ∗dN +NdN ,

and, if we assume NH = HR,

2dfH = dN −N ∗ dN . (1.13)

Let L = f∗T is the line bundle of the immersion f : M → H, and e =(

10

)a point at

infinity not intersecting f , i.e.,V = L⊕ eH .

An adapted complex structure S induces a holomorphic structure J on eH = V/L via thissplitting. In particular, S is given in the splitting V = L⊕ eH by

S =(J B

0 J

), (1.14)

where ∗δ = Jδ = δJ and B ∈ Γ(Hom(eH, L)). Since S2 = −1, we have

JB +BJ = 0 .


Note that if ψ =(f1

)∈ Γ(L) then

∗δ = Jδ = δJ ⇐⇒ ∗df = Ndf = −dfR ,

where N,R : M → S2 are defined by Jψ = −ψR and Je = eN .

The trivial connection d on V = L⊕ eH is given by

d =(∇L 0δ ∇

).

Since d is flat, both connections ∇L on L and ∇ on eH = V/L are flat, and

d∇δ = 0 ,

where ∇ is the induced connection on Hom(L, eH). Since the mean curvature spherecongruence has to satisfy a second order condition, we compute

dS =(∇LJ −Bδ ∇B

0 δB + ∇J

).

Here ∇ is the connection on Hom(eH, L) induced by ∇L and ∇. The (1, 0) and (0, 1)–partof dS with respect to S are given by

(dS)′ =12

(dS − S ∗ dS) =(

(∇LJ)′ (∇B)′ − 12(B ∗ δB +B ∗ ∇J)

0 δB + (∇J)′

)and

(dS)′′ =12

(dS + S ∗ dS) =(

(∇LJ)′′ −Bδ (∇B)′′ + 12(B ∗ δB +B ∗ ∇J)

0 (∇J)′′

)respectively. We will show that the mean curvature sphere congruence S of f satisfies

Be = −ψH (1.15)

where H = −HN is given by the mean curvature vector of f . Writing Be = −ψB withB : M → H we obtain

2(dS)′′ψ = 2(∇LJ)′ψ − 2Bδψ = ψ(2Bdf − (dR−R ∗ dR)) . (1.16)

Thus, B = −HN if and only if (dS)′′|L = 0.

We apply this computation to an embedded round sphere f : S2 → S4 in S4. The meancurvature sphere congruence S of f is constant (given by S(p) = f(S2) for all p ∈ S2).Thus dS = 0 and the above computation shows that S is given by (1.14) with Bψ = −ψHwhere H = HN and H is the mean curvature vector of f .


In particular, an adapted complex structure S is the mean curvature sphere congruence off , i.e., the mean curvature vectors H of S(p) at p and H of f at p coincide for all p ∈M ,if and only if (dS)′′|L = 0.

Since S2 = −1 the derivative of S anticommutes with S. Decomposing the derivative ofS into (1, 0) and (0, 1)–parts

dS = 2(∗Q− ∗A) ∈ Ω1(End−(V )) , (1.17)

where A ∈ Γ(K End−(V )) and Q ∈ Γ(K End−(V )), we see that a complex structureS ∈ Γ(End(V )), S2 = −1, is the mean curvature sphere congruence of f if and only if

∗ δ = Sδ = δS , and Q|L = 0 . (1.18)

Since S2 = −1 we have NH = HR, so that a similar computation, using (1.13), showsthat (1.18) is equivalent to

∗δ = Sδ = δS , and AV ⊂ L .

For latter use, we give the Hopf fields A and Q of an adapted complex structure in thesplitting V = L⊕ eH

A =12

(∗dS)′ =(AL 1

2(∗∇B)′ + 14(BδB +B∇J)

0 A+ 12 ∗ δB

)(1.19)

and

Q = −12

(∗dS)′′ =(QL + 1

2B ∗ δ −12(∗∇B)′′ + 1

4(BδB +B∇J)0 Q

)(1.20)

where (∇LJ) = 2(∗QL − ∗AL) and ∇J = 2(∗Q− ∗A) are the splittings of the derivativesof J and J into (0, 1) and (1, 0) parts respectively. In particular, S is the mean curvaturesphere congurence of f if and only if

A = −12∗ δB (1.21)

or, equivalently,

QL = −12B ∗ δ . (1.22)

The Willmore functional of a conformal immersion f : M → S4 from a compact Riemannsurface into the 4–sphere is given [Wil93] by

W(f) =∫M

(|H|2 −K −K⊥)|df |2 . (1.23)


HereH is the mean curvature vector of f , K the Gaussian curvature, andK⊥ the curvatureof the normal bundle. In our notation, the Willmore integrand computes [BFL+02, Prop.11] to

|H|2 −K −K⊥ =14|dR+R ∗ dR|2 .

Let < B >:= Re tr(B) then (1.19) gives

< A ∧ ∗A >= −2Re trA2 =18|dR+R ∗ dR|2

and the Willmore energy of a conformal immersion f : M → S4 can be expressed as

W(f) = 2∫< A ∧ ∗A > . (1.24)


For convenience, we collect how properties of conformal maps are phrased in the quater-nionic formalism.

Geometric property Quaternionic formulation

f : M → S4 map quaternionic line bundle L→ML ⊂ V = H2

df : TM → TS4 δ ∈ Ω1(Hom(L, V/L)) derivative of L

f : M → S4 holomorphic there exists J ∈ Γ(End(L)),J2 = −1, ∗δ = δJ

round 2–sphere S in S4 S ∈ End(H2), S2 = −1

point [x] lies on sphere S S[x] = [x]

sphere congruence S S ∈ Γ(End(H2)), S2 = −1

sphere congruence S envelopes f : M → S4 S is adapted: ∗δ = Sδ = δS

f : M → S4 conformal immersion δ is nowhere vanishing, and there existsan adapted complex structure

S ∈ Γ(End(V ))

Hopf fields A,Q A = 12(∗dS)′, Q = −1

2(∗dS)′′

S mean curvature sphere congruence S adapted, Q′′|L = 0of f (S conformal Gauss map of f)

Willmore functional W(f) = 2∫< A ∧ ∗A >,

W(f) =∫

(|H|2 −K −K⊥)|df |2 S conformal Gauss map of f


1.1.2 Holomorphic curves in HPn

In the previous section, we discussed holomorphic maps f : M → S4 = HP1 from aRiemann surface into the 4–sphere. In analogy to the theory of complex curves f : M →CPn, we extend our definition of holomorphicity to maps f : M → HPn from a Riemannsurface M into quaternionic projective space.

Consider maps f : M → Gk+1(V ) from a Riemann surfaceM into the (k+1)–Grassmannianof the trivial Hn+1 bundle V over M . Let T → Gk+1(V ) be the tautological (k+ 1)–planebundle whose fiber over Vk ∈ Gk+1(Hn+1) is TVk = Vk ⊂ V . A map f : M → Gk+1(V ) canbe identified with a rank k+ 1 subbundle Vk ⊂ V via Vk = f∗T , i.e., (Vk)p = Tf(p) = f(p)for p ∈ M . From now on, we will make no distinction between a map f into the Grass-mannian Gk+1(V ) and the corresponding subbundle Vk ⊂ V .

The derivative of Vk ⊂ V is given by the Hom(Vk, V/Vk) valued 1–form

δk = πVkd|Vk , (1.25)

where πVk : V → V/Vk is the canonical projection and d is the trivial connection on V .Under the identification TGk+1(V ) = Hom(T , V/T ) the 1–form δk is the derivative df off : M → Gk+1(V ).

Analog to the case n = 1, i.e. conformal maps f : M → S4 = HP1, we define a holomorphiccurve into a Grassmannian:

Definition 1.8. Let V = Hn+1 be the trivial quaternionic (n + 1)–plane bundle over aRiemann surface M and d the trivial connection on V .

A map f : M → Gk+1(V ) from a Riemann surface M into the Grassmannian Gk+1(V ) iscalled a holomorphic curve in Gk+1(V ) if there exists a complex structure J ∈ Γ(End(Vk)),J2 = −1, such that

∗δk = δk J ,

where Vk = f∗T and δk = πd|Vk is the derivative of Vk.

A holomorphic curve f : M → Gk+1(V ) is called full if Vk is not contained in a lowerdimensional Grassmannian.

We are particularly interested in the case when f : M → HPn is a holomorphic curve intoquaternionic projective space. We usually denote by L = f∗T the associated line bundle.If f = [f1, . . . , fn, 1] : M → HPn is given in affine coordinates then

δψ = π

df1

...dfn0

for ψ =

f1

...fn1

∈ Γ(L) .


If we define R : M → S2 by Jψ = −ψR, then f is a holomorphic curve with ∗δ = δJ ifand only if [fi, 1] : M → HP1 are conformal maps to S4, all of them having the same rightnormal R.

Example 1.9. A good example to keep in mind are the holomorphic curves in HPn whicharise as the twistor projections of smooth curves in CP2n+1. We view Hn+1 = C2n+2 viathe complex structure I given by right multiplication by i. The twistor projection mapsa complex line in Hn+1 into the corresponding quaternionic line via

C2n+2 ⊃ E 7→ π(E) = E ⊕ Ej ⊂ Hn+1 .

If h : M → CP2n+1 is a smooth curve with corresponding line bundle E ⊂ C2n+2 givenby Ep = h(p), then its twistor projection

L = π(E) = E ⊕ Ej ⊂ V = M ×Hn+1

is a (quaternionic) holomorphic curve if and only if the (0, 1)–part of the derivative δE =πEd|E of E with respect to the complex structure I on Hn+1 satisfies

δ′′E ∈ Γ(K HomC(E,L/E)) . (1.26)

To see this, define a complex structure J on L by regarding E as the +i eigenspace of J .For ϕ ∈ Γ(E) we have Jϕ = ϕi, and thus

δ′′Eϕ =12πE(dϕ+ (∗dϕ)i) =

12πE(dϕ+ ∗d(Jϕ)) .

In particular, (1.26) holds if and only

πL(Dψ) = 0

for ψ ∈ Γ(L) where D is the mixed structure

D =12

(d+ ∗dJ) (1.27)

on L. Since the derivative of f is given by δ = πL(d|L), we obtain

πL(Dψ) = δψ + ∗δJψ

for ψ ∈ Γ(L) so that (1.26) is equivalent to

∗δψ = δJψ ,

which means that L is a holomorphic curve.

In particular, if h : M → CPn is a complex holomorphic curve in CPn, i.e.,

δ′′E = 0 ,


Figure 1.3: Twistor projection of a holomorphic curve in CP3

then its twistor projection is a holomorphic curve in HPn.

Conversely, if f : M → HPn is a holomorphic curve, then the complex structure J on Ldefines a splitting

L = E ⊕ Ej

where E and Ej are the ±i eigenspaces of J respectively. The complex line bundle Egives a smooth map h : M → CP2n+1, the twistor lift of f .

Definition 1.10. The Frenet flag of a full holomorphic curve f : M → HPn is a full flagV0 = L ⊂ V1 ⊂ . . . ⊂ Vn−1 ⊂ Vn = V of quaternionic subbundles of rank Vk = k + 1together with complex structures Jk on the quotient bundles Vk/Vk−1, J0 = J , such that

1. dΓ(Vk) ⊂ Ω1(Vk+1).

2. The derivatives δk = πVkd : Vk/Vk−1 → T ∗M ⊗ Vk+1/Vk satisfy

∗δk = Jk+1δk = δkJk .

Note that the Frenet flag exists smoothly away from isolated points on M , the so–calledWeierstraß points of f , and extends continuously into the Weierstraß points [FLPP01,Lemma 4.10]. In case the Frenet flag extends smoothly into the Weierstraß points, theseare exactly the zeros of the derivatives δk. In particular, a conformal immersion f : M →S4 has no Weierstraß points.

Example 1.11. Consider the line bundle L = f∗T ⊂ H2 induced by the holomorphicmap f : M → S4 = HP1. In this case, the flag bundles exist globally and are given byL ⊂ V1 = H2. This flag is a Frenet flag if there exists J on V/L with ∗δ = Jδ. In otherwords, [f, 1] : M → HP1 is a holomorphic curve with Frenet flag if and only if there existleft and right normals N,R : M → S2 with ∗df = Ndf = −dfR.

Similar to the case of a holomorphic curve in S4 we call a complex structure adapted if itinduces the complex structures given by the Frenet flag:

Definition 1.12. Let f : M → HPn be a holomorphic curve with Frenet flag L ⊂ V1 ⊂. . . ⊂ V . A complex structure S ∈ Γ(End(V )) is called adapted if

∗δk = Sδk = δkS ,

where δk : Vk → V/Vk are the derivatives of the flag bundles Vk.

Lemma 1.13. A holomorphic curve f : M → HPn has a Frenet flag if and only if thereexists a smooth adapted complex structure on V .


Proof. The existence of S on M renders δ0 into a complex holomorphic section, [Les]. Inparticular, the zeros fo δ0 are isolated and the image of δ0 defines a smooth bundle. Bydefining inductively the flag bundles as the smooth bundles given by the derivatives δk,one shows that the Frenet flag exists smoothly on M , see [LP03]. For the converse, choosea splitting V = L⊕ L1 ⊕ . . .⊕ Ln such that the flag spaces of f are given by

Vk =k⊕i=0

Li ,

and define a complex structure S by

S =

J1 0. . .

0 Jn

,

where Ji are the induced complex structures on Li when identifying Li = Vi/Vi−1 via thesplitting. By definition, S is an adapted complex structure.

Remark 1.14. The derivative δ0 of a holomorphic curve f : M → HP1 = S4 which hasa smooth adapted complex structure and df 6≡ 0, has isolated zeros [Les]. Therefore, amap f : M → R4 with smooth (N,R) : M → S2 × S2 satisfying ∗df = Ndf = −dfR is abranched conformal immersion.

In particular, a holomorphic map f : M → R3 is a branched conformal immersion sincethe left normal equals (1.4) the right normal N : M → S2 given by the holomorphicitycondition.

Remark 1.15. As in the case n = 1, an adapted complex structure can be seen as acongruence of osculating CPn’s: Let E ⊂ Hn+1 be a complex line subbundle of Hn+1 =C2n+2 such that E ⊕ Ej = Hn+1 where we consider Hn+1 = C2n+2 via the complexstructure I given by right multiplication by i. The map

E ⊃ wC 7→ wH ⊂ Hn+1

is injective and defines a CPn ⊂ HPn. On the other hand, splittings E ⊕ Ej = Hn+1 arethe same as complex structures S ∈ End(Hn+1), S2 = −1, by declaring E to be the +ieigenspace of S. The condition that the congruence of CPn passes through a holomorphiccurve f is given by

SL = L ,

where L = f∗T is the line bundle of f . Since the tangent space to CPn at Lp is given by

TLpCPn = Hom+(L,Hn+1/L)p ,

the congruence of CPn’s envelopes f if and only if

δ ∈ Γ(K Hom+(L,Hn+1/L)) .


We decompose the derivative of a complex structure S ∈ Γ(End(V )), S2 = −1, into (1, 0)and (0, 1)–parts with respect to S:

dS = 2(∗Q− ∗A) ,

where A ∈ Γ(K End−(V )) and Q ∈ Γ(K End−(V )). Given a holomorphic curve f : M →HPn and a hyperplane at infinity α ∈ (Hn+1)∗ such that V = L ⊕ kerα, an adaptedcomplex structure is given in this splitting as

S =(J B

0 S

)where B ∈ Γ(Hom(kerα,L)), S ∈ Γ(End(kerα)) is a complex structure on kerα, and

JB +BS = 0 .

Since S is adapted we also have∗δ = Sδ = δJ .

The same computation as in the 1–dimensional case (1.19), (1.20) gives the formulas forthe Hopf fields A and Q of an adapted complex structure S in the splitting V = L⊕kerα:

A =12

(∗dS)′ =(AL 1

2(∗∇B)′ + 14(BδB +B∇J)

0 A+ 12 ∗ δB

)(1.28)

and

Q = −12

(∗dS)′′ =(QL + 1

2B ∗ δ −12(∗∇B)′′ + 1

4(BδB +B∇J)0 Q

). (1.29)

Example 1.16. Let h : M → CP2n+1 be a smooth curve such that its twistor projectionf : M → HPn is a holomorphic curve (1.26), i.e.,

δ′′E ∈ Γ(K HomC(E,L/E)) .

If S is a complex structure on V which induces J on L, i.e.,

S|L = J ,

thenδ′′E = A|E .

This follows from the fact that L/E = Ej so that the projection

πE : L→ L/E


is the projection onto the −i eigenspace of J = S|L. Therefore, we have for ϕ ∈ Γ(E):

δ′′Eϕ =14

((dϕ+ ∗d(Sϕ) + Sd(Sϕ)− S ∗ dϕ)|L = Aϕ .

In particular, f : M → HPn is the twistor projection of a holomorphic curve h : M → CPnin CPn, i.e., δ′′E = 0, if and only if

A|L = 0 (1.30)

for any complex structure S with S|L = J .

Moreover, if the nth flag space En of the holomorphic curve h : M → CPn does not containa quaternionic subspace, i.e.,

En ⊕ Enj = C2n+2 = Hn+1

then En defines a complex structure S on Hn+1. The flag spaces Vk = Ek ⊕Ekj of L areS–stable and δk|Ek = δEk shows that S is adapted.

Example 1.17. In what follows, the example of the dual curve of a holomorphic curvef : M → HPn with Frenet flag will play an important role. For any subbundle Vk of V let

V ⊥k := α ∈ V ∗ |< α,ψ >= 0 for all ψ ∈ Vk ,

where V ∗ is the dual bundle of V .

The dual curve L† of a holomorphic curve L ⊂ V with Frenet flag is the holomorphiccurve in V ∗ defined by

L† := V ⊥n−1 .

In the case of a holomorphic curve f : M → S4 with adapted complex structure, the dualcurve is given by the antipodal map since L† = V ⊥n−1 = L⊥. The Frenet flag of the dualcurve L† of L is given by

V †k = V ⊥n−1−k . (1.31)

with derivatives δ†k = −δ∗n−1−k and complex structures J†k = J∗n−k. In particular, if S isan adapted complex structure of L then S∗ is an adapted complex structure of the dualcurve L†. Since

(dS)∗ = d∗S∗

where d∗ is the dual connection of d, one easily verifies that the Hopf fields of f † are givenby

A† = −Q∗ ∈ Γ(K End−(V ∗)) and Q† = −A∗ ∈ Γ(K End−(V ∗)) . (1.32)


Example 1.18. We use the above concept of the dual curve to define a holomorphiccurve in HPn by prescribing an n–dimensional subbundle of V . Let f : M → HPn be aholomorphic curve in HPn and ω be a nowhere vanishing closed 1–form with imω = Lwhere L = f∗T so that kerω is a n–dimensional subbundle of V . Moreover, assume thatω ∈ Γ(K Hom(V,L)) where the complex structure J on L is given by the holomorphiccurve f , i.e.,

∗ω = Jω .

For ψ ∈ Γ(kerω) we see

0 = d(ωψ) = (dω)ψ − ω ∧ dψ = −ω ∧ πkerωdψ ,

where πkerω : V → V/ kerω is the canonical projection. Therefore, the derivative δω =πkerωd|kerω of kerω satisfies

∗δω = Jδω ,

where J is the complex structure on V/ kerω induced by the bundle isomorphism ω :V/ kerω → L. Since L = (kerω)⊥ has derivative δ = −δ∗ω, we see that

∗δ = δ∗ ,

in other words, L = (kerω)⊥ is a holomorphic curve. Note that L is contained in aconstant subbundle V ⊂ V ∗ if and only if V ⊥ ⊂ kerω. In particular, f is a full curve inHPn if and only if kerω does not contain a constant subbundle.

As a consequence we can give a holomorphic curve by prescribing the nth flag space:

Corollary 1.19. Let ω ∈ Γ(K Hom(V,L)) be a nowhere vanishing 1–form with dω = 0such that kerω does not contain a constant subbundle. Then a holomorphic curve f :M → HPn with flag bundles Vk is the dual curve of L = (kerω)⊥ if and only if

Vn−1 = kerω .

We discuss the existence of ω ∈ Γ(K Hom(V,L)) wit dω = 0 in Chapter 2.

1.1.3 Frenet curves in HPn

The analog of the conformal Gauss map of a conformal immersion f : M → S4 is thecanonical complex structure of a holomorphic curve in HPn. Recall that in the case n = 1,the conformal Gauss map corresponds to an adapted complex structure S with Q|L = 0.Similarly, we define:

Definition 1.20. Let f : M → HPn be a holomorphic curve with Frenet flag L ⊂ V1 ⊂. . . ⊂ V .


An adapted complex structure S ∈ Γ(End(V )) with

Q|Vn−1 = 0 (1.33)

is called the canonical complex structure of f . Here

dS = 2(∗Q− ∗A) (1.34)

is the decomposition of the derivative of S into (1, 0) and (0, 1)–parts A ∈ Γ(K End−(V ))and Q ∈ Γ(K End−(V )) respectively.

Lemma 1.21 ([Les]). Let Vk ⊂ V be stable with respect to a complex structure S ∈Γ(End(V )) on V , and A and Q the Hopf fields of S. Then ∗δk = Sδk = δkS implies thatVk is A and Q stable. Here δk = πVkd|Vk is the derivative of Vk.

Using the above result we show that Q|Vn−1 = 0 is equivalent to AV ⊂ Ω1(L):

Corollary 1.22. An adapted complex structure S ∈ Γ(End(V )) is the canonical complexstructure of f if and only if

AV ⊂ Ω1(L) . (1.35)

Proof. If Q|Vn−1 = 0 then (1.34) yields

12πLd(Q+A)|Vk =

12πLd(S(∗Q− ∗A))|Vk = πL(∗Q− ∗A) ∧ (∗Q− ∗A)|Vk

= πL(Q ∧Q+A ∧A)|Vk = πLA ∧A|Vk , (1.36)

where we used Q|Vn−1 = 0 and A ∧ Q = Q ∧ A = 0 by type considerations. SincedQ|Vk = Q ∧ δk = 0 by type (1.36) becomes

πLdA|Vk = πLA ∧A|Vk .

Because A stabilizes the flag spaces Vk for all k, we obtain πLA ∧A|L = 0 and

0 = πLdA|L = δ0 ∧A|L + πLA ∧ δ0 = πLA ∧ δ0

shows that AV1 ⊂ Ω1(L). Proceeding inductively, we conclude AV ⊂ Ω1(L).

Conversely, if AV ⊂ Ω1(L) we use the dual curve L† to show that Q|Vn−1 = 0: SinceQ† = −A∗, we see that

L ⊃ imA = (kerA∗)⊥ = (kerQ†)⊥

which shows that V †n−1 = L⊥ ⊂ kerQ†. By the previous argument this implies

V ⊥n−1 = L† ⊃ imA† = (kerA∗)⊥ = (kerQ)⊥

so thatVn−1 ⊂ kerQ .


The proof of the previous Corollary is exemplary for latter arguments: using the dual curvewe transfer properties of the Hopf field Q to properties of A. In particular, a complexstructure S is the canonical complex structure of f if and only if S∗ is the canonicalcomplex structure of its dual curve f †.

Example 1.23. Let f : M → HPn be the twistor projection of a holomorphic curveh : M → CP2n+1, and assume that the nth–flag space of h does not contain a quaternionicsubspace, i.e,

En ⊕ Enj = Hn+1 .

As discussed before, En induces an adapted complex structure S on V . By the samearguments as in Example 1.16, we see that

0 = δ′′Ek = A|Ek

since Ek is the +i eigenspace of the complex structure S restricted to the kth flag spaceVk = Ek ⊕ Ekj of f . Thus,

A = 0 ,

and S is the canonical complex structure of f .

In general, the Frenet flag and the canonical complex structure of a holomorphic curvef : M → HPn only exist [FLPP01, Lemma 4.1] away from the Weierstraß points of f .Whereas the Frenet flag of a holomorphic curve extends continuously into the Weierstraßpoints [FLPP01, Lemma 4.10], the canonical complex structure may become singular: Iff : M → HP1 is the twistor projection of a complex holomorphic curve h : M → CP3

then the tangent W1 ⊂ V of h can become quaternionic, i.e., W1 = W1j at some p ∈ M ,see [Pet04]. In this case the canonical complex structure S degenerates to a point atp ∈M and thus S cannot be extended into p ∈M . To avoid these difficulties, we considerholomorphic curves f : M → HPn which have a smooth canonical complex structure. Forconformal maps f : M → HP1 this means that the mean curvature sphere congruenceextends smoothly across the branch points.

Definition 1.24. A Frenet curve f : M → HPn is a holomorphic curve which has asmooth canonical complex structure on M .

Recalling Lemma 1.13 we have

Corollary 1.25 ([LP03]). The Frenet flag of a Frenet curve f : M → HPn is smooth onM .

A trivial example of a Frenet curve is an unramified curve f : M → HPn, that is aholomorphic curve without Weierstraß points. In the case n = 1 an unramified curve isa conformal immersion f : M → S4 so that the conformality gives an adapted complexstructure on V , and we can solve (1.21) smoothly for B. Moreover, the dual curve f † ofa Frenet curve f : M → HPn is Frenet.


Definition 1.26. The Willmore energy of a Frenet curve f : M → HPn over a compactRiemann surface M is given by

W(f) = 2∫M< A ∧ ∗A > , (1.37)

where A is the Hopf field of the canonical complex structure S of f .

Remark 1.27. As we will see below, the definition of the Willmore energy is a Mobiusinvariant. In the case n = 1, we obtain (1.24) the usual Willmore functional (1.23)

W(f) =∫M

(|H|2 −K −K⊥)|df |2 .


Again, we summarize:


f : M → HPn map line bundle L→M , L ⊂ V = Hn+1

df : TM → THPn δ ∈ Ω1(Hom(L, V/L))

f : M → HPn holomorphic there exists J ∈ Γ(End(L)), J2 = −1,∗δ = δJ

Frenet flag of f L = V0 ⊂ V1 ⊂ . . . ⊂ Vn = V withJk ∈ Γ(End(Vk/Vk−1)), J2

k = −1, such that

dΓ(Vk) ⊂ Ω1(Vk+1)and

∗δk = δkJk = Jk+1δk

S adapted complex structure S ∈ Γ(End(V )), S2 = −1,and ∗δk = Sδk = δkS

f † dual curve of f L† = V ⊥n−1

Hopf fields of S A = 12(∗dS)′, Q = −1

2(∗dS)′′

S canonical complex structure S adapted, Q|Vn−1 = 0

f Frenet curve f has smooth canonical complex structure

Willmore energy of a Frenet curve W(f) = 2∫< A ∧ ∗A >

1.2. HOLOMORPHIC LINE BUNDLES 35

1.2 Holomorphic line bundles

In this section we briefly explain the notion of a (quaternionic) holomorphic structure on aquaternionic vector bundle, and how holomorphic curves f : M → HPn are in one-to-onecorrespondence to holomorphic line bundles [FLPP01]. Analytically, the Cauchy–Riemannoperator ∂ on a complex vector bundle is replaced in the quaternionic theory by an ellipticoperator D = ∂+Q on a complex quaternionic vector bundle (W,J), where Q is a (0, 1)–form with values in the complex antilinear endomorphisms of W and ∂ is a complexholomorphic structure on the complex bundle (W,J). The quaternionic holomorphic linebundle (W,D) inherits a new invariant, the Willmore energy W(W,D) = 2

∫M < Q∧∗Q >

of the holomorphic structure D. The properties of quaternionic holomorphic structures arein many ways similar to the properties of complex holomorphic structures, for examplethe vanishing orders of holomorphic sections are well–defined, and the Riemann–Rochtheorem holds verbatim. The quaternionic Plucker formula involves the new invariant,and gives eventually estimates on the Willmore energy.

1.2.1 Holomorphic vector bundles

In what follows, let (W,J) be a quaternionic vector bundle with complex structure J .

Definition 1.28. A quaternionic holomorphic structure on (W,J) is given by a quater-nionic linear operator

D : Γ(W )→ Γ(KW ) (1.38)

satisfying the Leibniz rule D(ψλ) = (Dψ)λ + (ψdλ)′′, where ψ ∈ Γ(W ) and λ : M → H.The quaternionic vector space of holomorphic sections of W is denoted by

H0(W ) = ker(D) (1.39)

and has finite dimension h0(W ) := dimH0(W ) for compact M . A linear subspace H ⊂H0(W ) is called a linear system.

The zeros of a holomorphic section ψ ∈ H0(W ) are isolated, and the vanishing orderordp ψ of ψ at a zero p ∈ M is well–defined [FLPP01, Def. 3.5]. The Weierstraß gapsequence of H is given by n0(p) < . . . < nn(p) where n0(p) is the minimal vanishing orderof a holomorphic section in H at p, and each nk(p) is defined inductively by letting nk(p)be the minimal vanishing order of holomorphic sections in H at p strictly greater thannk−1(p) for k ≥ 1.

The order of a linear system H is defined [FLPP01, Def. 4.2] by

ord(H) =∑p∈M

ordp(H) , (1.40)


where ordp(H) =∑n

k=0(nk(p)− k) is the order of H at p and n0(p) < . . . < nn(p) is theWeierstraß gap sequence of H. Away from isolated points the Weierstraß gap sequence isnk(p) = k and ordpH =

∑n−1k=0 nk − k measures the deviation from the generic sequence.

A quaternionic vector bundle W with complex structure J over a Riemann surface Mdecomposes into W = W+ ⊕W−, where W± are the ±i–eigenspaces of J . By restrictionJ induces complex structures on W± and W− = W+j gives a complex linear isomorphismbetween W+ and W−. The decomposition of D into J commuting and anticommutingparts gives

D = ∂ +Q . (1.41)

Here ∂ = ∂ ⊕ ∂ is the double of a complex holomorphic structure on W+ and Q ∈Γ(K End−(W )) is a (0, 1)–form with values in J–complex J–antilinear endomorphismsof W . The endomorphism Q ∈ Γ(K End(V )) “measures” the difference to the complextheory of complex holomorphic structures ∂ ⊕ ∂ on E ⊕ E.

Definition 1.29. The L2–norm

W(W ) =W(W,D) = 2∫M< Q ∧ ∗Q > (1.42)

of Q is called the Willmore energy of the holomorphic bundle (W,D) where < , > denotesthe trace pairing on End(W ). The special case Q = 0, for which W(W ) = 0, describes(doubles of) complex holomorphic bundles W = W+ ⊕W+.

A connection ∇ on a complex quaternionic vector bundle (W,J) decomposes

∇ = ∇+ +∇− (1.43)

into J–commuting and J–anticommuting parts, in particular ∇ = ∇+ is a complex con-nection on W . Furthermore, we denote the (0, 1) and (1, 0)–parts of ∇+ and ∇− by

∇ = ∂ + ∂ and ∇− = Q+A ,

where∗∂ = −S∂ = −∂S , ∗∂ = S∂ = ∂S ,

and Q ∈ Γ(K End−(W )) and A ∈ Γ(K End−(W )). Note that

∇J = 2(∗Q− ∗A)

since ∇J = 0. The (0, 1)–part of ∇ defines a holomorphic structure

∇′′ = ∂ +Q

on (W,J). Moreover, if ∇ is a flat connection then

0 = R+ d∇(Q+A) +Q ∧Q+A ∧A ,


where we used that A ∧ Q = Q ∧ A = 0 by type considerations. Splitting this equationinto J commuting and J–anticommmuting parts, we get

R = −(Q ∧Q+A ∧A) , (1.44)

andd∇(Q+A) = 0 .

We denote by W the complex vector bundle (W,−J) of the complex quaternionic vectorbundle (W,J). Then ∇′ is the (0, 1)–part of the connection ∇ with respect to −J and(W ,∇′) is a holomorphic vector bundle. We call ∇′ an anti–holomorphic structure on(W,J).

The degree of the quaternionic bundle W with complex structure J is defined as the degreeof the underlying complex vector bundle

degW := degW+ , (1.45)

which is half of the usual degree of W when viewed as a complex bundle with J . Thedegree of a complex bundle is given by the curvature of a complex connection on W .Therefore, if ∇ = ∇+A+Q is a flat connection on (W,J) then (1.44) gives

degW =1

2π

∫M< JR >=

12π

∫M< A ∧ ∗A > − < Q ∧ ∗Q > . (1.46)

Given two quaternionic bundles W and W with complex structures J and J the complexlinear homomorphisms Hom+(W, W ) are complex linearly isomorphic to HomC(W+, W+),in particular

deg Hom+(W, W ) = deg W − degW . (1.47)

On the other hand, the complex antilinear homomorphisms Hom−(W, W ) are complexlinearly isomorphic to Hom+(W , W ), where the complex structure on a homomorphismbundle is induced by the target complex structure.

Finally, if V1 and V2 are two complex holomorphic vector bundles with complex holo-morphic structures ∂k, then Hom+(V1, V2) inherits a complex holomorphic structure ∂via

(∂A)ψ := ∂2(Aψ)−A(∂1ψ) . (1.48)

The usual tensor product construction for complex holomorphic structures induces a com-plex holomorphic structure on K Hom+(V1, V2).


We summarize:


complex quaternionic vector bundle quaternionic vector bundle W with

complex structure J ∈ Γ(End(W ))

degree of a complex degW = degW+

quaternionic vector bundle (W,J) where W+ is the +i eigenspace of J

holomorphic structure D on (W,J) D = ∂ +Q, Q ∈ Γ(K End−(W ))

complex holomorphic structure D = ∂

Willmore energy of (W,D) W(W,D) = 2∫< Q ∧ ∗Q >, D = ∂ +Q

space of holomorphic sections of (W,D) H0(W ) = kerD

linear system of holomorphic sections H ⊂ H0(W ) linear subspace

order of a linear system H at p ∈M ordpH =∑

(nk(p)− k)

where nk(p) is the Weierstraß gapsequence

1.2.2 The Kodaira correspondence

We now turn to the question, how holomorphic curves f : M → HPn correspond toholomorphic line bundles (W,D). In the case of complex holomorphic curves f : M →CPn the Kodaira correspondence gives a 1:1 correspondence between holomorphic curvesand holomorphic line bundles with basepoint free linear systems H. In fact, given aholomorphic curve f : M → CPn the corresponding holomorphic line bundle is given by


E−1 where E is the line bundle associated to the holomorphic curve via Ep = f(p). As inthe case of algebraic curves in CPn, the canonically associated holomorphic line bundle ofa holomorphic curve f : M → HPn is given by the dual bundle L−1 of L = f∗T via theKodaira correspondence:

A holomorphic curve f : M → HPn induces, [FLPP01, Thm. 2.3], a unique holomorphicstructure D on the dual bundle L−1 of L = f∗T such that linear forms α ∈ V ∗ on Vrestricted to L give holomorphic sections

α|L ∈ H0(L−1) .

Thus H = V ∗ ⊂ H0(L−1) is a (n + 1)–dimensional linear system. By transversality His basepoint free, that is, there exists a nowhere vanishing holomorphic section of theholomorphic line bundle L−1 in H.

Conversely, a holomorphic line bundle L−1 and a basepoint free (n+1)–dimensional linearsystem H ⊂ H0(L−1) give a holomorphic curve f : M → HPn = P(H∗) by the Kodairaembedding of L with respect to the linear system H which is defined as

ev∗(L) ⊂ V = (M ×H)∗ .

Here the bundle mapev : M ×H → L−1, (p, ψ) 7→ ψp

evaluates the holomorphic section ψ at the point p.

Theorem 1.30 (Kodaira correspondence, see [FLPP01, Thm. 2.8]). There is a 1:1 cor-respondence between holomorphic curves f : M → HPn (up to Mobius equivalence) andholomorphic line bundles L−1 together with basepoint free linear systems H ⊂ H0(L−1).

In particular, every holomorphic curve f : M → HPn gives rise to a family of conformalmaps into S4: choosing a basepoint free 2–dimensional linear system H ⊂ H yields viathe Kodaira embedding of L ⊂ V = (H)∗ a conformal map f : M → S4.

If we choose α ∈ H0(L−1), α|L 6= 0, then there is a unique ψ ∈ Γ(L) with < α,ψ >= 1.We define R : M → S2 by Jψ = −ψR so that for all β ∈ H0(L−1) the map

fβ =< β,ψ >: M → H

is conformal with right normal vector R since the Leibniz rule for the holomorphic struc-ture D gives

0 = Dβ = α(dfβ +R ∗ dfβ) .

In other words, if we fix a point at infinity α, then every 2–dimensional linear systemH ⊂ H0(L−1) with α ∈ H gives rise to a conformal map f : M → H with right normal R.

Conversely, every conformal map f : M → H with right normal R defines a holomorphicsection βf = αf ∈ H0(L−1). This way, we can think of a holomorphic curve f : M → HPnwith dimH0(L−1) = n+ 1 as the family of all conformal maps f : M → H with the sameright normal vector.


Example 1.31. Consider the dual curve f † : M → HPn of a holomorphic f : M → HPnwith Frenet flag. Then L† = V ⊥n−1 and (L†)−1 = V/Vn−1. Since

(πn−1d|Vn−1)′ = δ′n−1 = 0 ,

where πn−1 : V → V/Vn−1 is the canonical projection, the operator D : Γ((L†)−1) →Γ(K(L†)−1) given by

D[α] = (πn−1dα)′′ (1.49)

for α ∈ Γ(V ) is well–defined. One easily verifies that D is a holomorphic structure onV/Vn−1. In fact, we obtain the holomorphic structure given by the Kodaira correspondenceon V/Vn−1 = (L†)−1.

Remark 1.32. Note that the notion of the dual curve depends on the linear system off : M → HPn = P (H∗): if L ⊂ H∗ has dual curve L† ⊂ H and H & H is basepoint free,then in general

π(L)† 6= π(L†) ,

where π : H∗ → H∗ denotes the canonical projection. This is due to the fact that ifdimH = n + 1 then the n − 1 first derivative of L ⊂ H occurs in the dual curve L†

whereas in π(L)† only lower derivatives appear.

Since a holomorphic curve f : M → HPn corresponds to a holomorphic line bundle L−1,we can associate the Willmore energy to a holomorphic curve:

Definition 1.33. Let f : M → HPn be a holomorphic curve. The Willmore energy of fis defined by

W(f) :=W(L−1, D) =∫M< Q ∧ ∗Q >

where D = ∂ +Q is the induced holomorphic structure of L−1.

Note that the Willmore energy is by definition a Mobius invariant since it only dependson the holomorphic structure D on L−1.

Example 1.34. A holomorphic curve f : M → HPn is the twistor projection of a holo-morphic curve h : M → CPn if and only if (1.30)

AL = 0

where AL is the Hopf field of the complex structure J on L given by the holomorphicity off . Since in this case the holomorphic structure on L−1 is given by ∂+Q where Q∗ = −AL,we see that twistor projections of holomorphic curves in CPn have zero Willmore energy.

Conversely, if f : M → HPn has zero Willmore energy then 0 = Q∗ = −AL, that is, thecomplex bundle E given by the +i eigenspace of J on L has δ′′E = AL|E = 0. This showsthat f is the twistor projection of the holomorphic curve h : M → CP2n+1 given by thecomplex bundle E.


For a Frenet curve f : M → HPn with canonical complex structure S, the complexstructure on L−1 is given by (S|L)∗ = J , and the Hopf fields are related by

(A|L)∗ = −QL−1,

where A is the Hopf field with respect to S whereas QL−1

is given by the (0, 1)–part of J .In particular, the Willmore energy is given by (1.37) in case of a Frenet curve

W(f) = 2∫< Q ∧ ∗Q >= 2

∫< (A|L)∗ ∧ (∗A|L)∗ >= 2

∫< A ∧ ∗A > .

The holomorphic structure ∂ of a complex holomorphic vector bundle E−1 induces a holo-morphic structure ∂∗ on E by dualization. For quaternionic line bundles this dualizationprocess fails: the dual D∗ on V ∗ of a quaternionic holomorphic structure D on V is amixed structure [FLPP01, Lemma 2.1] but not a holomorphic structure.

To equip the line bundle L of a holomorphic curve f : M → HPn with a holomorphicstructure, one can choose a hyperplane at infinity not intersecting f : the basepoint freelinear system H has a nowhere vanishing holomorphic section and thus we can chooseα ∈ V ∗ so that V = L⊕ kerα. In this splitting we decompose the trivial connection d onV as

d|L = ∇L + δ . (1.50)

Here δ is the derivative of f expressed via the identification V/L = kerα and the inducedconnection ∇L on L is flat since kerα ⊂ V is constant. We now equip the line bundle Lwith the holomorphic structure

D = (∇L)′′ (1.51)

coming from the connection ∇L on L. Note that L always has the canonical nowherevanishing holomorphic section ψ ∈ H0(L) for which < α,ψ >= 1: since ∇Lψ = 0 thesection ψ also has Dψ = 0 and thus is holomorphic. In particular, the complete linearsystem H0(L) is basepoint free. However, the holomorphic structure D on L dependson the choice of α. We will return in Chapter 2 to the holomorphic curves given by theKodaira embedding L−1 ⊂ (H0(L))∗.

1.2.3 The Riemann–Roch Theorem

In what follows, we frequently will make use of the bundle KL of L–valued (1, 0)–forms.By [PP98] there is a unique holomorphic structure DKL on KL compatible with thepairing

L−1 ×KL→ Λ2TM∗ ⊗H :

given ψ ∈ Γ(L−1) and ω ∈ Γ(KL) the holomorphic structure is determined by the Leibnizrule

d < ψ, ω >=< Dψ ∧ ω > + < ψ,DKLω > . (1.52)


The Riemann–Roch theorem relates the dimensions of the spaces of holomorphic sectionsof L−1 and KL. It is verbatim the same statement as in the complex case. In fact,since the index of the elliptic operator D = ∂ + Q does not change under continuousdeformations, we have index(∂ + tQ) = index(∂) for t ∈ [0, 1]. The holomorphic structureDKL on KL is the adjoint elliptic operator to D [FLPP01, Thm. 2.2] so that

index(D) = dimH0(L−1)− dimH0(KL) .

Using the Riemann–Roch theorem for complex holomorphic vector bundles we see:

Theorem 1.35 (Riemann–Roch Theorem, see [FLPP01, Thm. 2.2]). Let L−1 be a holo-morphic line bundle over a Riemann surface M and KL be equipped with the holomorphicstructure so that L−1 and KL are paired holomorphic bundles. Then

dimH0(L−1)− dimH0(KL) = degL−1 − g + 1 , (1.53)

where g is the genus of M .

Two descriptions of the holomorphic structure DKL on KL will be useful for our purposes:first, since L ⊂ V is a subbundle, we can take exterior derivatives of ω ∈ Γ(KL) withrespect to the trivial connection d on V . But πdω = δ ∧ ω = 0 by type considerationswhich implies that dω ∈ Ω2(L) is again L–valued. One immediately checks (1.52) for dand thus

DKL = d . (1.54)

Second, if ∇L−1is any connection on L−1 adapted to the holomorphic structure D on

L−1, i.e., D = (∇L−1)′′, then the exterior derivative of ω ∈ Γ(KL) with respect to the

dual connection ∇L on L also gives

DKL = d∇L. (1.55)

This follows again from the fact that d∇L

satisfies (1.52).

Lemma 1.36. Let f : M → HPn be a holomorphic curve with f = [f1, . . . , fn, 1] in affinecoordinates, and denote by

ψ =

f1

...fn1

∈ Γ(L) .

Then ω ∈ H0(KL) if and only if there exists locally a map g : U ⊂M → H such that

ω = ψdg and dfk ∧ dg = 0

for some 1 ≤ k ≤ n.


Proof. Since f is holomorphic, we know that ∗δ = δJ for some complex structure J on L.Let R : M → S2 be defined by Jψ = −ψR, then ∗δψ = δJψ is equivalent to ∗dfi = −dfiRfor all i. In particular, dfk ∧ dg = 0 for some 1 ≤ k ≤ n implies dfk ∧ dg = 0 for all1 ≤ i ≤ n.

Moreover, a 1–form ω ∈ Γ(KL) is d∇L–closed if and only if there exists locally a map gwith ω = ψdg. In this case ∗ω = Jω is equivalent to

0 = δ ∧ ω = δψ ∧ dg = π

df1 ∧ dg

...dfn ∧ dg

0

.

1.2.4 The Plucker formula

The Plucker formula [FLPP01, Thm. 4.7] gives a lower bound on the Willmore energyW(D) of a holomorphic bundle (L,D) over a compact Riemann surface M in terms ofthe genus of M , the degree of L, and the vanishing orders of holomorphic sections of L.We give a proof of the Plucker formula in the case when L is the holomorphic line bundlegiven by a Frenet curve f : M → HPn.

In the first step, we compute the degrees of the flag spaces of f .

Lemma 1.37. Let f : M → HPn be a Frenet curve, S its canonical complex structureand L ⊂ V1 ⊂ . . . ⊂ Vn = V its Frenet flag with corresponding derivatives δk. Then thedegree of the bundle Vk/Vk−1 with respect to S is given by

deg Vk/Vk−1 =k−1∑i=0

ord δi − k degK + degL, 0 ≤ k ≤ n . (1.56)

where we put V−1 := 0.

Proof. The degree of a complex holomorphic line bundle E is given by the vanishing orderof any holomorphic section of E. In [Les] it is shown that the derivatives of the flag spacesare holomorphic sections

δi ∈ H0(K Hom+(Vi/Vi−1, Vi+1/Vi))

provided there exists an adapted complex structure S on V . Here the complex holomorphicstructure on K Hom+(Vi/Vi−1, Vi+1/Vi) is given by the tensor construction (1.48) and the


complex holomorphic structures ∂ = d′′+ on Vi/Vi−1 and Vi+1/Vi. Therefore, (1.47) showsthat

ord δi = deg(K Hom+(Vi/Vi−1, Vi+1/Vi)) = degK + deg Vi+1/Vi − deg Vi/Vi−1 .

Telescoping this identity, we get

k−1∑i=0

ord δi = k degK + deg Vk/Vk−1 − degL . (1.57)

Remark 1.38. The degree of the dual curve f † of a Frenet curve f : M → HPn is given by

degL† = n degK − degL−n−1∑i=0

ord δi , (1.58)

since (V/Vn−1)−1 = V ⊥n−1 = L†.

We now prove the quaternionic Plucker relation [FLPP01, Thm. 4.7] in the case of aFrenet curve:

Theorem 1.39 (Plucker formula). Let f : M → HPn be a Frenet curve and S its canonicalcomplex structure. Let L ⊂ V1 ⊂ . . . ⊂ Vn = V be the Frenet flag of f and δk thederivatives of Vk. If M is a compact Riemann surface of genus g, then

deg(V, S) =1

4π(W(f)−W(f †)) = (n+ 1)(n(1− g) + degL) + ordH , (1.59)

where ordH is the order of the linear system H = V ∗ ⊂ H0(L−1).

Remark 1.40. The expression (1.40) for the order of H simplifies for a Frenet curve f :M → HPn to

ordH =n−1∑i=0

(n− i) ord δi . (1.60)

Proof. Since V =⊕n

k=0 Vk/Vk−1 as complex vector bundles, we obtain the right handsight of the equation by using (1.57)

deg(V, S) =n∑k=0

deg Vk/Vk−1 =n∑k=0

(k−1∑i=0

ord δi − k degK + degL)

= (n∑k=0

k−1∑i=0

ord δi)−n(n+ 1)

2degK + (n+ 1) degL .


On the other hand, recalling (1.46)

2π deg(V, S) =∫M< A ∧ ∗A > − < Q ∧ ∗Q >

as well as (1.37) and (1.32)

W(f) = 2∫M< A ∧ ∗A > , W(f †) = 2

∫M< Q ∧ ∗Q >

we get4π deg(V, S) =W(f)−W(f †) .

Remark 1.41. Let h : M → CP2n+1 be a holomorphic curve in CPn such that the twistorprojection f : M → HPn of h is a Frenet curve, see Example 1.23. Since W(f) = 0 thePlucker relation shows that the Willmore energy of the dual curve f † of f is given by

W(f †) = 4π deg(V, S) ∈ 4πN .

The general case of the Plucker formula is proved [FLPP01, Thm. 4.7] by using theholomorphic jet complex of the holomorphic line bundle L−1. Away from the Weierstraßpoints p1, . . . , pm, the kth jet bundle is given by the dual Lk = V ∗k of the flag spaceVk. Moroever there exists an isomorphism P : H → Ln of the basepoint free linearsystem H ⊂ H0(L−1) and the nth jet space of L−1 on M0 = M \ p1, . . . , pm. Thetrivial connection on H can be pushed forward to a trivial connection ∇ on Ln over M0.Moreover, the holomorphic jet complex Ln has a canonical complex structure, which isgiven on M0 by the dual of the canonical complex structure S = S∗ of f . Decomposing∇ with respect to the complex structure S we get

∇ = ∇+A+Q ,

where ∇ is a S–complex connection with only logarithmic singularities at the Weierstraßpoints [FLPP01, Sec. 4.3], and∫

M< SR >= 2π(degLn − ordH) .

Recall (1.44) thatR = −(A ∧A+Q∧Q)

and W(L) = 2∫< Q∧ ∗Q >, which shows that

W(L†) :=∫< A ∧ ∗A >

is finite. Computing the degree of the nth–jet bundle, one derives the Plucker formula inthe general case. For the complete argument see [FLPP01, Sec. 4.3].


Discussing the orders of zeros of holomorphic sections carefully [FLPP01, Sec. 4.5] thePlucker formula gives estimates on the Willmore energy of a conformal map f : M → S4

in dependence of the dimension of the space of holomorphic sections, e.g., in the case ofa conformal torus with dimH0(L−1) = n+ 1 one has [FLPP01, Rem. 6]

W(f) ≥

π(n+ 1)2 , n oddπ((n+ 1)2 − 1) , n even

.

Using these kinds of estimates, one derives a new estimate for the eigenvalues of the Diracoperator of a Riemannian spin bundle [FLPP01, Thm. 5.5], area estimates for a constantmean curvature torus in terms of its spectral genus g [FLPP01, Thm. 6.5]

area(f) ≥

π4 (g + 2)2 , g evenπ4 ((g+)2 − 1) , g odd

,

and estimates on the Willmore energy of a constant mean curvature torus in R3 or S3

[FLPP01, Thm. 6.7]

W(f) ≥

π4 (g + 2)2 , g evenπ4 ((g+)2 − 1) , g odd

.

In particular, to verify the Willmore conjecture on minimal tori in the 3–sphere it sufficesto consider minimal tori of spectral genus at most 3.

The case of equality in the Plucker formula are the so–called soliton surfaces, which arediscussed for example in [Pet04], [BP].

Chapter 2

Transformations on conformalmaps

Transformations which preserve special surface classes in 3– and 4–space play an impor-tant role in surface geometry. One of the motivations to study those transformationscomes from the fact that they allow to construct more complicated surfaces from givensimple ones. Historical examples include the Backlund transformation on surfaces of con-stant Gaussian curvature [Bia80] and the Darboux transformation on isothermic surfaces[Dar99]. More recently, also a Backlund transformation and a Darboux transformationon Willmore surfaces in S4 have been studied [BFL+02].

In this chapter we discuss a Darboux and a Backlund transformation on conformal mapsf : M → S4 of a Riemann surface M into the 4–sphere [BLPP] [LP05], [Pet04], [BP].Both transformations satisfy Bianchi permutability: The Darboux transformation arisesfrom integrating a Ricatti–type equation whereas the Backlund transformation comes fromsolving Abelian integrals (which explains to some extend our choice of notation).

2.1 The Darboux transformation

The classical Darboux transformation of isothermic surfaces goes back to [Dar99]. Ge-ometrically, the Darboux transform of an isothermic surface f : M → S4 is given by asphere congruence enveloping both f and its Darboux transform f ]. To define a Darbouxtransformation on arbitrary conformal maps, we relax the enveloping condition [BLPP].We show that the transformation defined this way satisfies Bianchi permutability. In par-ticular, we can define the spectral curve of a conformal torus f : T 2 → S4 in S4 as theset of all Darboux transforms: For each point p ∈ T 2, the images f ](p) of the Darbouxtransforms f ] of f canonically embed the spectral curve into S4 as a twistor projection ofa holomorphic curve F (p, .) : Σ→ CP3.

47

48 CHAPTER 2. TRANSFORMATIONS ON CONFORMAL MAPS

In the special case of a constant mean curvature torus f : T 2 → R3 this definition of thespectral curve coincides [CLP] with the “classical” one given by the eigenvalues of theholonomy of a family of flat connections [Hit90]. We show that the Darboux transformscorresponding to points on the spectral curve are isothermic even though the generalDarboux transformation only coincides for very special points on the spectral curve withthe classical Darboux transformation of a constant mean curvature torus in R3.

2.1.1 The classical Darboux transformation on isothermic surfaces

The classical Darboux transformation of isothermic surfaces in 3–space goes back to[Dar99]. We recall the notion of an isothermic surface f : M → S4, see for example[HJ02, Sec. 7.4], [Bur]:

Definition 2.1. A conformal map f : M → S4 is called isothermic if there exists a closed1–form non–trivial ω ∈ Ω1(R), i.e.,

dω = 0 and imω ⊂ L ⊂ kerω . (2.1)

In this case, any local parallel section ψ of d+ ω gives rise to a quaternionic line bundleL = ψH over an open set U of M . The corresponding map f : U ⊂ M → S4 is called aclassical Darboux transform of f .

In [HJ02, Sec. 7.4] it is shown that the 1–form ω ∈ Ω1(R) is unique up to multiplicationby a real constant ρ ∈ R, at least away from the umbilics of f . The condition (2.1) impliesthat d + ωρ is a flat connection on V for all ρ ∈ R. We fix ω ∈ Ω1(R) for reference andchoose a point at infinity not intersecting f then ω is written in the splitting L⊕ eH = Vas

ω =(

0 η0 0

).

Note that ω is closed if and only

0 = dω =(−η ∧ δ d∇η

0 δ ∧ η

). (2.2)

If ψ ∈ Γ(pr∗ V ) is a parallel section of d+ ωρ, ρ ∈ R, on the universal cover pr : M →Mof M then

ψ =(α+ fββ

)with dα = −dfβ since

dψ = −ωψρ ∈ Ω1(pr∗ L) . (2.3)

Note thatdα = −dfβ

2.1. THE DARBOUX TRANSFORMATION 49

shows that γ∗ψ = ψhγ for all γ ∈ π1(M) with hγ : M → H∗, so that the classical Darbouxtransform L = ψH is globally defined on M . Moreover,

T = αβ−1

is a global solution of the Ricatti equation

dT = −df + T ηρT , (2.4)

where ηρ = −dβα−1. Note that (2.3) shows

ωe = ηe =(f1

)η ,

and ω is closed (2.2) if and only if

dη = 0 , df ∧ η = 0 , and η ∧ df = 0 (2.5)

Using dα = −dfβ one easily verifies

(dη)T = −η ∧ df and df ∧ η = 0 ,

so that we proved:

Corollary 2.2. If d+ ω is a connection with ω ∈ Ω1(R) and if there exists

ψ =(α+ fββ

)∈ Γ(pr∗ V )

with dψ = −ωψ, then ω is closed if and only if dβα−1 is closed.

Figure 2.1: Darboux pair

Definition 2.3. Let f and f be conformal maps into the 4–sphere such that L ⊕ L =V . The pair (f, f) is called a (classical) Darboux pair if both surfaces f and f aresimultaneously enveloped by a sphere congruence S.

Recall (1.9) that S envelopes f if and only if

∗δ = Sδ = δS .

We write the trivial connection d and the sphere congruence S in the splitting V = L⊕ L:

d =(∇L δ

δ ∇

)(2.6)


and

S =(J 00 J

), (2.7)

where we used SL = L and SL = L since S envelopes f and f . Moroever, the envelopingcondition

∗δ = Jδ = δJ and ∗ δ = Jδ = δJ

is equivalent toδ ∧ δ = 0 and δ ∧ δ = 0 . (2.8)

An isothermic immersion f : M → S4 and each of its classical Darboux transformsf : M → S4 form a Darboux pair (f, f). Since the argument is of local nature we mayassume that ψ ∈ Γ(L) is a parallel section of d + ω where ω ∈ Ω1(R) is closed. Sincedψ = −ωψ ∈ Ω1(L) we get, using the splitting L⊕ L = V ,

ωψ = −δψ .

For ψ ∈ Γ(L) we have0 = (dω)ψ = d(ωψ) + ω(dψ)

so that0 = πLd(ωψ) = δ ∧ ωψ = −δ ∧ δψ

and∗ δ = Jδ , (2.9)

where J is the complex structure on V/L = L given by the conformality of f . Similarly,for ψ ∈ Γ(L) we have

0 = (dω)ψ − ω ∧ πLdψ = −ω ∧ δψ ,

i.e., using δ = −ω,∗ δ = δJ , (2.10)

where J is the complex structure on V/L = L given by f . In particular, the classicalDarboux transform f : M → S4 of an isothermic surface f is conformal. Moreover, wecan define a sphere congruence S on V = L⊕ L by

S|L = J and S|L = J

and S envelopes f and f by (2.10) and (2.9).

Conversely, the surfaces f and f of a Darboux pair are isothermic: If we define

ω =(

0 δ0 0

)∈ Ω1(R)

in the splitting V = L⊕ L then ω is closed since

dω =(−δ ∧ δ d∇δ

0 δ ∧ δ

)= 0 ,


where we used (2.8) and d∇δ = 0 by the flatness of d. Interchanging the roles of L and Lwe see that f is isothermic, too.

One of the important features of the Darboux transformation on isothermic surfaces isthe Bianchi permutability , see for example [HJ02, Sec. 7.6]:

Theorem 2.4 (Bianchi permutability). Given two Darboux transforms f1, f2 : M → S4

of an isothermic surface f : M → S4 then there exists an isothermic surface f : M → S4

such that f is simultaneously Darboux transform of f1 and f2.

Proof. Let ρ1, ρ2 ∈ R such that Li is d + ωρi parallel, and let ψi ∈ Γ(pr∗ Li) be parallelsections with respect to d+ωρi. Since dψi = −ωρiψi ∈ Γ(pr∗KL), there exists ρ : M → Hsuch that

dψ2 = dψ1ρ .

Define f : M → S4 by L = ψH where

ψ = ψ2 − ψ1ρ .

To see that f is a Darboux transform of f1, we write

ψi =(αi + fβi

βi

)in the splitting L⊕ eH = V so that

ωρie =(f1

)dβiα

−1i

andρ = (ρ1α1)−1(ρ2α2) .

The Darboux transforms are given in affine coordinates by

[fi, 1] : M → S4

where fi = f +Ti and Ti = αiβ−1i satisfy the Ricatti equation (2.4). Therefore, ψ is given

in the splitting L1 ⊕ eH = V as

ψ =(α+ f1β

β

)where

β = β2 − β1ρ = ((ρ2T2)−1 − (ρ1T1)−1)ρ2α2

andα = (T2 − T1)β2 .

Using the Ricatti equation (2.4) and dα2 = −dfβ2 we obtain

dβ = −(ρ1T1)−1dfT−11 ρ2α ,


and(dβ)α−1 = −(ρ1T1)−1dfT−1

1 ρ2 .

Applying again the Ricatti equation this yields

d((dβ)α−1) = −((ρ1T1)−1dfT−11 + η) ∧ dfT−1

1 ρ2 + (ρ1T1)−1df ∧ (T−11 dfT−1

1 − ηρ1)ρ2

= 0 ,

where we also used that η ∧ df = df ∧ η = 0 by (2.5). Corollary 2.2 shows that f is aclassical Darboux transform of f1. Interchanging the roles of f1 and f2 we see that f andf2 form a Darboux pair, too.

If (f, f) form a Darboux pair then f is a Darboux transform of f and vice versa. If wechoose the reference ω ∈ Ω1(R) according to this symmetry, the parameters in the Bianchipermutability are given by the following diagram, see [HJ02, Sec. 7.6]:

f1

ρ2

%%KKKKKKKKKKKKK

f

ρ1

99sssssssssssss

ρ2

%%KKKKKKKKKKKKK f

f2

ρ1

99sssssssssssss

Note also that f can be computed in terms of f1 and f2 by only using differentiationand algebraic operation. For more results on isothermic surfaces see for example [Bur],[HJP97], [HJ02], [HJ97], [KPP98].

2.1.2 The spectral curve of constant mean curvature tori

To motivate how we generalize the classical Darboux transformation on isothermic surfacesto a Darboux transformation on conformal maps f : M → S4, we first explain how pointson the spectral curve of a constant mean curvature torus f : T 2 → R3 can be seen asconformal maps which satisfy a generalized enveloping condition.

Let f : M → R3 = im H be a constant mean curvature surface, and without loss ofgenerality assume that f has constant mean curvature H = −1. Thus, the mean curvaturesphere congruence of f is given in the splitting V = L⊕ eH by (1.14) and (1.15)

S =(J R

0 J

)with Re = ψ. Using the mean curvature sphere condition (1.21)

∗A =12δR


and(∇R)e = ∇L(Re)−R(∇e) = ∇Lψ = 0 ,

we get

d∇ ∗ A = (12d∇δ)R− 1

2δ∇R = 0 , (2.11)

where we used that d∇δ = 0 by the flatness of d. Thus ∗A is a closed 1–form whichimplies by standard arguments, see for example [BFL+02, Prop. 5], that J is harmonic.Let λ = a+ bJ , a, b ∈ R, |λ| = 1, then

∇λ = ∇+ (λ− 1)A (2.12)

defines a family of connections on eH. Note that

∇λ = b(∇J) = 2b(∗Q− ∗A)

and0 = d∇ ∗ A = d∇(JA) = J(d∇A− 2A ∧ A)

since Q ∧ A = 0 by type considerations.

This gives for ωλ = (λ− 1)A

d∇ωλ + ωλ ∧ ωλ = 2b(∗Q− ∗A) ∧ A+ 2(λ− 1)A ∧ A+ |λ− 1|2A ∧ A = 0 ,

where we used that |λ| = 1. Thus, the connection ∇λ = ∇+ωλ is flat. The family ∇λ of flatquaternionic connections extends to a family of flat complex connections [FLPP01, Lemma6.3] if we introduce the complex structure I on V/L = eH given by right multiplicationby i: For µ = α+ βI ∈ C∗ the connections

∇µ = ∇+µ+ µ−1 − 2

2A+

µ−1 − µ2

I ∗ A

are flat by the same computation as before since a = µ+µ−1

2 and b = µ−1−µ2 satisfy

a2 − b2 = 1.

Fixing a base point p ∈ M , we get a family of holonomy representations Hµ : π1(M) →SL(2,C) for µ ∈ C∗. We now restrict to the case of a torus M = T 2 so that the funda-mental group π1(M) is abelian and we have common eigenvectors for the holonomy of ∇µ.The set of eigenvalues of Hµ compactifies to a hyperelliptic Riemann surface Σ → CP1,the so–called spectral curve of f , of finite genus [Hit90]. The eigenlines of Hµ define aholomorphic line bundle L → Σ over the spectral curve. Note that these considerationswere done with respect to a fixed base point p ∈M . When changing p the spectral curveΣ remains unchanged however the point in the Jacobian corresponding to L changes lin-early with the base point on (the abelian group) T 2. For details how to reconstruct theharmonic map N : T 2 → S2 and thus the constant mean curvature torus from algebraiccurve data compare [Hit90], [McI01].


In order to deal with a quaternionic connection instead of a complex one, introduce [CLP]the complex structure Sx on V/L = eH as the quaternionic extension of the complexstructure I on the complex Eµ bundle to the quaternionic bundle Eµ ⊕ Eµj = eH anddefine

∇x = ∇+ Aµ+ µ−1 − 2

2+ ∗ASxµ

−1 − µ2

, (2.13)

where we denote in abuse of notation µ = α + βSx. Then ∇x is a flat quaternionicconnection on eH with

∇xSx = 0

and∇x|Γ(Eµ) = ∇µ|Γ(Eµ) .

Moreover, for µ(x) ∈ S1 the connection ∇x is the quaternionic connection defined in(2.12).

Lemma 2.5 ([CLP]). A point x ∈ Σ on the spectral curve gives a unique pair (∇x, Sx)on eH with

∇x = ∇ − A+ Aa+ ∗ASxb , a2 − b2 = 1 ,

and ∇xSx = 0.

We want to understand points on the spectral curve geometrically. Let ϕ ∈ Γ(pr∗ V/L)be a parallel section with respect to

∇x = ∇+ ω

on the universal cover pr : M → M of M . Here we again identify V/L = eH andabbreviate

ω = Aµ+ µ−1 − 2

2+ ∗ASxµ

−1 − µ2

.

For a lift ψ ∈ Γ(pr∗ V ) of ϕ, i.e.,πL(ψ) = ϕ ,

where πL : V → V/L is the canonical projection, we write

ψ = ϕL + ϕ

in the splitting V = L⊕ eH. There is a well–defined map B ∈ Hom(eH, L) given by

δB = ω (2.14)

since δX : L→ eH is for X 6= 0 a bundle isomorphism and ∗ω = Jω and ∗δ = Jδ. Becausedψ = ∇Lϕl + δϕL − ωϕ, we see that

πL(dψ) = 0 if and only if ϕL = Bϕ .

We summarize:


Lemma 2.6. Let ϕ ∈ Γ(pr∗ V/L) be a parallel section of a connection ∇ + ω on V/Lwhere ∇ is the trivial connection on V/L = eH and ω a 1–form with values in V/L with∗ω = Jω. Then there is a unique lift ψ ∈ Γ(pr∗ V ) of ϕ so that

πL(dψ) = 0 .

Note, that ψ has the same (multiplicative) monodromy as ϕ, and ψ is nowhere vanishingsince πL(ψ) = ϕ is a parallel section of ∇µ. Therefore, ψ defines a line subbundle of V by

Lµ = ψH ⊂ V ,

and V splits into the direct sumV = L⊕ Lµ .

Writing the trivial connection d in this splitting, i.e., identifying V/L = Lµ and V/Lµ = L,we have

d =(∇L δµ

δ ∇µ)

Note that ∇µ is indeed the flat connection ∇x if we identify eH = V/L = Lµ by ϕ 7→ ψsince ∇µψ = πLdψ = 0. Together with the flatness of d we obtain

0 = Rµ = −δ ∧ δµ .

Therefore, the map fµ : T 2 → S4 given by Lµ is conformal since

∗δµ = Jδµ ,

where J is the complex structure on V/Lµ given by the identification V/Lµ = L and theconformality of f .

We call fµ a (generalized) Darboux transform of f : the sphere congruence given by S|L = Jand S|Lµ = J envelopes f and left–envelopes fµ, that is SLµ = Lµ and

∗δµ = Jδµ .

We have shown:

Lemma 2.7. If f : M → R3 is a constant mean curvature torus with spectral curve Σthen points x ∈ Σ give (generalized) Darboux transforms fµ of f .

2.1.3 The Darboux transformation on conformal maps into the 4–sphere

As we have seen in the previous section, points on the spectral curve of a constant meancurvature torus f : T 2 → R3 give (generalized) Darboux transforms f : T 2 → S4. TheDarboux transformation can be defined for conformal maps f : M → S4, and gives


eventually rise to the spectral curve of a conformal torus f : T 2 → S4 in the 4–sphere,[BLPP].

The Darboux transformation is defined by relaxing the enveloping condition: a spherecongruence left envelopes a conformal map f ] : M → S4 if S(p) goes through f ](p) andthe tangent planes of f and S(p) are left–parallel for all p ∈M . Two planes in R4 throughthe origin are called left–parallel, if their oriented intersection great circles in S3 ⊂ R4

correspond via right translation in the group S3.

Definition 2.8 ([BLPP]). Let f : M → S4 be a conformal map. If there exists a spherecongruence S and a conformal map f ] such that L ⊕ L] = V and S envelopes f andleft–envelopes f ] then f ] is called a Darboux transform of f .

Figure 2.2: Homogeneous torus with various Darboux transforms

Note that the sphere congruence S is in this case given in the splitting V = L⊕ L] as

S =(J 00 J ]

).

Moreover, writing the trivial connection d =(∇L δ]

δ ∇])

as before in the splitting, the

touching and left touching condition read as

∗δ = J ]δ = δJ and ∗ δ] = Jδ] .

In other words, f ] is a Darboux transform of f if and only if

δ ∧ δ] = 0 . (2.15)

The flatness of d implies that (2.15) is equivalent to the flatness of the connection ∇] onL]. Since

d|Γ(L]) = ∇] + δ]

a section ψ] ∈ Γ(pr∗ L]) satisfies

dψ] ∈ Ω1(pr∗ L)

if and only if∇]ψ] = 0 .

We summarize:

Corollary 2.9 ([BLPP]). Let f : M → S4 be a conformal immersion and f ] : M → S4

such that V = L⊕ L]. The following statements are equivalent:


1. f ] is a Darboux transform of f .

2. ∇] is a flat connection on L].

3. There exists a non–trivial section ψ] ∈ Γ(pr∗ L]) with monodromy such that dψ] ∈Ω1(pr∗ L).

Note that(∇])′′ = D = (πd)′′

is the holomorphic structure (1.49) on V/L induced by the dual surface f † of f , and isthus independent of f ]. Moreover, parallel sections of ∇] give holomorphic sections of D.

Conversely, a holomorphic section ϕ] ∈ H0(pr∗ V/L) with monodromy has a unique liftψ] ∈ pr∗ V with the same monodromy and πLdψ

] = 0: note that in the proof of Lemma2.6 we only used the fact that

(∇ϕ])′′ = 0 ,

where ∇ is the trivial connection on V/L induced by d and the splitting V = L ⊕ eH.But (ϕ])′′ = 0is satisfied in our situation since for any lift ψ = ϕ] + ϕ]L of a holomorphicsection ϕ] ∈ H0(pr∗ V/L) we have

(πL(dψ))′′ = Dϕ] = 0 .

If this unique lift ψ] of a holomorphic section ϕ ∈ H0(pr∗ V/L) is nowhere vanishing thenψ]H = L] defines a conformal map f ], and f ] is a Darboux transform of f . More general,we call the map f ] which is defined away from the zeros of ψ] a singular Darboux transform.

Figure 2.3: Darboux transforms of a homogeneous torus

Note that the holomorphicity conditions is equivalent to a Ricatti type equation: if ϕ ∈H0(pr∗ V/L) is a holomorphic section with monodromy, and f ] a Darboux transform off , then the Ricatti–type equation

ω = −(∇B) +BδB (2.16)

for B ∈ Hom(eH, L) where ∗ω = Jω has a solution since there exists a unique lift ψ] ∈Γ(pr∗ V ) of ϕ with πLdψ

] = 0. Conversely, if B is a solution of this equation for some ω


with ∗ω = Jω then ∇+ δB is a flat connection on V/L since

d∇(δB) + δB ∧ δB = δ ∧ ω = 0 .

Any parallel section ϕ ∈ Γ(pr∗ V/L) of ∇+ δB is holomorphic in V/L since (∇+ δB)′′ =∇′′ = D, in other words,

(ϕ+Bϕ)H

is a Darboux transform of f .

Remark 2.10. Note that a connection d+ ω such that L] is d+ ω parallel and ω ∈ Ω1(R)is uniquely given by (2.16). Moreover, as we have seen in Section 2.1, f and f ] form aclassical Darboux pair if and only if dω = 0.

As in the case of the Darboux transformation of isothermic surfaces, we have Bianchipermutability:

Theorem 2.11 ([BLPP]). Given two Darboux transforms f ] and f [ of a conformal mapf : M → S4 then there is a common Darboux transform f of f ] and f [.

Proof. Let ϕ], ϕ[ ∈ H0(pr∗ V/L) be the holomorphic sections with monodromy, such thatf ] and f [ are given by the unique lifts ψ], ψ[ ∈ Γ(pr∗ V ) of ϕ] and ϕ[ respectively withπL(dψ]) = πL(dψ[) = 0. In other words,

dψ] = dψ[ρ ,

where ρ : M → H. Define, as in the case of isothermic surfaces,

ψ = ψ[ − ϕ]ρ ,

then dψ ∈ Ω1(pr∗ L]) so thatL = ψH

is a Darboux transform of f ]. Similarly, ψρ−1 exhibits L as a Darboux transform off [.

2.1.4 The Darboux transformation on constant mean curvature surfaces

It is a well–known fact that a constant mean curvature surface f : M → R3 (without lossof generality with H = −1) is isothermic: Recalling (1.13) we see

(dN)′ = −df and (dN)′′ = dg ,

where g = f +N : M → im H . In particular,

df ∧ dg = dg ∧ df = 0


so that

ω =(

0 dg0 0

)∈ Ω1(R)

is a closed 1–form. We fix ω as the reference 1–form used in Section 2.1.

The Darboux transformation is even in this case a genuine generalization of the classicalDarboux transformation: only for special parameters x ∈ Σ of the spectral curve Σ of aconstant mean curvature torus we obtain a sphere congruence enveloping both surfaces fand fµ where µ = µ(x).

Lemma 2.12. Let f : T 2 → R3 be a constant mean curvature torus in R3. Let

∇x = ∇+ A(a− 1 + JSxb)

be the flat connection (2.13) given by a point x ∈ Σ of the spectral curve Σ of f whereµ = µ(x) and a = µ+µ−1

2 and b = µ−1−µ2 . The Darboux transform fµ given by ∇x is a

classical Darboux transform of f if and only µ ∈ R or µ ∈ S1.

Proof. Let ϕ be a parallel section of ∇x, and ψ its unique lift to V such that πL(dψ) = 0.The Darboux transform of f given by ∇x is given by Lµ = ψH.

Writing ψ = ϕ+Bϕ with B ∈ Hom(eH, L) and

ω =(

0 −∇B +BδB0 0

)then d+ ω is the unique connection with (d+ ω)ψ = 0 and ω ∈ Ω1(L). Recall (2.14) that

δB = ωx = A(a− 1 + JSxb) .

Since the connection ∇x = ∇+ ωx is flat, we have

0 = dwx + ωx ∧ ωx = δ ∧ (−∇B +BδB) ,

so thatd∇(−∇B +BδB) = (∇B −BδB) ∧ δB .

Because

dω =(

(−∇B +BδB) ∧ δ d∇(−∇B +BδB)0 δ ∧ (−∇B +BδB)

)it remains with Remark 2.10 to show that

0 = (∇B −BδB) ∧ δ

if and only if µ ∈ R or µ ∈ S1.

We compute δ in the splitting L ⊕ eH = V . Let R : eH → L as before be defined byRe = ψ. Then R−1 is parallel with respect to the induced connections ∇L on L and ∇


on V/L, and R−1 is antilinear with respect to the complex structures on L and V/L, i.e.,R−1J = −JR−1. Since Je = eN we have

−4 ∗ Ae = (∇J − J ∗ ∇J)e = 2e(dN)′ = −2edf = −2δψ

so that

A(a− 1 + JSxb) = δB = −2AJR−1B ,

and

R−1B = Ja− 1

2− Sx b

2. (2.17)

Since µ is ∇x–parallel and a2 − b2 = 1, we get

R−1(∇B −BδB) = ∇(R−1B)−R−1BδB

= (∇xJ)a− 1

2− [δB,R−1B]−R−1BδB

= ∗Q(a− 1) + ∗A(

1− a− 12

(a− 1)2 +12b2)

= ∗Q(a− 1) .

But ∗Q(a− 1)∧ δ = 0 if and only if a commutes with J . Since a = µ+µ−1

2 this only holdsfor a ∈ R (since J = ±Sx is impossible as ∇xJ 6= 0), so that µ ∈ R or µ ∈ S1.

Remark 2.13. Implicitly, the above computation also gives the Ricatti equation

∇T = T ∗ AT (1− a)− 2 ∗ Q (2.18)

for T = 2R−1B(1− a)−1 .

Though the Darboux transformation is in general not the classical Darboux transformationon constant mean curvature tori, the Darboux transforms of a constant mean curvaturetorus f : T 2 → im H are still isothermic:

Theorem 2.14. Let f : T 2 → R3 be a constant mean curvature torus with spectral curveΣ. For every x ∈ Σ the corresponding Darboux transform fµ : M → S4 is isothermic.

Proof. To show that the Darboux transform fµ of f is isothermic it is enough to find aconformal map f so that f and fµ form a classical Darboux pair. Choose x2 ∈ Σ suchthat µ(x2) ∈ R (or S1), then the corresponding Darboux transform fµ(x2) is isothermic.Using Bianchi permutability there is a common Darboux transform f of fµ and fµ(x2).We show that f and fµ form a classical Darboux pair (whereas in general f and fµ(x2)


do not). For simplicity of notation we abbreviate f1 = fµ and f2 = fµ(x2).

f1

CDT

%%LLLLLLLLLLLLL

f

DT

99rrrrrrrrrrrrr

CDT

%%LLLLLLLLLLLLL f

f2

DT

99rrrrrrrrrrrrr

Let ϕi ∈ H0(pr∗ V/L) be the parallel section of

∇i = ∇+ A(ai − 1 + JSibo)

and ψi = ϕi + Biϕi ∈ Ω1(Li) the unique lift of ϕi with πLdψi = 0. We write ψi incoordinates

ψi =(αi + fβi

βi

),

where dαi = −dfβi. Moreover, we denote as before by fi = f + Ti with Ti = αiβ−1i .

Since

Bie =(f1

)T−1i ,

and2 ∗ Q(1− ai)e = edgλi ,

where λi = 1−ai2 in the trivialization V/L = eH and g = f +N , the Ricatti type equation

(2.16) reads in coordinates as

d(T−1) = T−1dfT−1 − dgλ (2.19)

On the other hand, the Ricatti equation (2.18) gives

d((λT )−1) = (λT )−1dfT−1 − dg . (2.20)

Note that the above two equations are equivalent if λ ∈ R but in general, i.e., for aconformal map f : M → HP1, the Ricatti type equation (2.16) does not imply (2.18).

The common Darboux transform f is given by

ψ =(α+ f1β

β

)where

β = ((λ2T2)−1 − (λ1T1)−1)λ2α2


andα = (T2 − T1)β2 .

Examining the computations in the proof of the Bianchi permutability Theorem 2.4 forisothermic surfaces one verifies that we only used λ2 ∈ R, df ∧ dg = dg ∧ df = 0, and theequations (2.19) and (2.20). Therefore, the same computation as before gives d(dβα−1) =0, so that f and f1 form a Darboux pair by Corollary 2.2.

2.1.5 The spectral curve of a conformal torus

Figure 2.4: Spectral curve of a homogeneous torus

Motivated by the definition of the spectral curve in case of a constant mean curvaturetorus, we expect the spectral curve of a conformal torus f : T 2 → S4 to basically be theset of all (singular) Darboux transforms of f . To turn this set into a Riemann surface[BLPP], we have to assume that f has trivial normal bundle. As we have seen, Darbouxtransforms of f are given by holomorphic setions ϕ ∈ Ho(pr∗ V/L) with monodromy. If

γ∗ϕ = ϕhγ

where h : Γ → H∗ is a representation of the fundamental group Γ of T 2 = C/Γ then thesection ϕ = ϕλ, λ ∈ H∗, has monodromy

γ∗(ϕ) = ϕλ−1hγλ .

In partiuclar, to represent a Darboux transform of f , we can choose ϕ ∈ H0(pr∗ V/L)such that hγ ∈ C∗ where C = Span1, i. We define

Spec(D) = h : Γ→ C∗ | there exists ϕ ∈ H0(pr∗ V/L) with γ∗ϕ = ϕhγ for γ ∈ Γ/C∗ .

Note that γ∗(ϕj) = ϕjhγ so that we have an involution τ(h) = h on Spec(D). It can beshown [BLPP] that Spec(D) defines a Riemann surface, the so–called spectral curve Σ off , of possibly infinite genus. In case of a constant mean curvature torus f : M → R3 itcoincides [CLP] with the spectral curve of [Hit90].


Figure 2.5: Spectral curve of a homogeneous torus, with original and Darboux transformedtorus, with Darboux transform

The space of holomorphic sections of pr∗ V/L with a given monodromy is generically 1–dimensional, and defines a complex holomorphic line bundle L → Σ over the spectralcurve which moves linearly in the Jacobian of Σ tangent to its Abel image due to Bianchipermutability [BLPP]. On the other hand, holomorphic sections ϕ] give singular Darbouxtransforms f ] of f . Thus, if we fix a point p ∈ T 2 on the torus and evaluate f ](p) for allpoints on the spectral curve we obtain a realization of the abstract Riemann surface Σ inS4.

If Σ is of finite type then the original surface gives a marked point ∞ ∈ Σ. Tracingthe marked point on the spectral curve under the flow of the holomorphic line bundle Lparametrizes the conformal torus f . In particular, we get a recipe to construct finite typeconformal immersions of 2–tori via theta functions on the spectral curve [BLPP].

2.1.6 The Darboux transformation on holomorphic curves

Similarly to the case of conformal maps f : M → S4 we can define a Darboux transfor-mation on holomorphic curves by giving holomorphic sections with monodromy.

Let f : M → HPn be an unramified Frenet curve. Its dual curve f † defines by Kodairacorrespondence a holomorphic structure D on V/Vn−1 = (L†)−1. The holomorphic jetcomplex [FLPP01, Thm. 4.3] of (L†)−1 is given by

Lk = V/Vn−1−k ,

with projections πk : V/Vn−1−k → V/Vn−k and Nk = kerπk = Vn−1−k/Vn−2−k. Inparticular, for a holomorphic section ϕ ∈ H0(pr∗ V/Vn−1) with monodromy there is aunique section ψ ∈ Γ(V ), the so–called nth prolongation of ϕ, such that for the successivelydefined sections ϕi+1 = πiϕi, ϕ0 = ψ, the following conditions hold [FLPP01, Corollary3.2]:

πidϕi = 0 , and ϕn = ϕ .

In particular, ψ has the same (multiplicative) monodromy as ϕ and we can define a mapf : M → HPn by

L = ψH .

We call f a Darboux transform of f .

If f : M → S4 is a conformal map such that dimH0(L−1) = n+ 1 > 2 then the Kodairaembedding of L ⊂ (H0(L−1))∗ gives a holomorphic curve f : M → HPn. The following


diagram

f : M → HPn

π

DT // f : M → HPn

π

f : M → HP1 DT // f : M → HP1

is commutative that is the projection of a Darboux transform f of f is a Darboux transformof the projection f of f . Little more is known about the Darboux transformation onholomorphic curves in HPn, in particular it is not yet understood how the existence ofmany holomorphic sections of a conformal torus f : T 2 → S4 affect the spectral curve off .

2.2 The Backlund transformation

A Backlund transformation is defined by solving Abelian Integrals [LP05], [BP]: Theholomorphic structure on KL is defined such that holomorphic sections in KL are exactlythe closed 1–forms of L. Integrating such a closed 1–forms ω ∈ H0(KL) on the universalcover pr : M →M of M , we get sections ϕ ∈ Γ(pr∗ L) with dϕ = ω. The line bundle pr∗ Lcan be equipped with a holomorphic structure D such that the ϕ’s are holomorphic, andspan a linear system H ⊂ H0(L). The Kodaira embedding of the line bundle L−1 ⊂ H∗

defines a holomorphic curve in HPk when dimH = k+1 ≥ 2. The Riemann Roch theoremgives control over the dimension of the space of holomorphic sections of KL, and thus ofdimH0(L).

More general, the choice of a hyperplane at infinity equips the line bundle L with aholomorphic structure (1.51). Any linear system H ⊂ H0(L) of dimension at least 2 givesby the Kodaira embedding of L−1 in H∗ a holomorphic curve, a Backlund transform of f .

Geometrically, the Backlund transformation is the quaternionic analog of a well–knowntransformation in algebraic curve theory: given a holomorphic curve in h : M → CPnthe intersection of the tangent of h with a fixed hyperplane gives a holomorphic curve inh : M → CPn−1, the osculate of h. Conversely, prescribing the tangents of h : M → CPn−1

one can define an envelope h : M → CPn such that h is an osculate of h [LP03]. Thisgeometric construction is closely related to the Backlund transformation: a Backlundtransform of f turns out to be a projection of an envelope of f .

Though a 1-step Backlund transform is in general only defined on the universal cover M ofM , we show that the (n+1)–step Backlund transform given by holomorphic ωi ∈ H0(KL)is given globally by differentiation and algebraic operations.

2.2. THE BACKLUND TRANSFORMATION 65

2.2.1 The Backlund transformation on holomorphic curves

As we have seen in (1.51) the choice of a hyperplane at infinity α ∈ (Hn+1)∗ not intersectingthe holomorphic curve f : M → HPn induces a holomorphic structure on L: if V =L⊕ kerα then the trivial connection d on L induces (1.50) a trivial connection ∇L on Land (1.51)

D = (∇L)′′

is the induced holomorphic structure on L. If H ⊂ H0(L) is a basepoint free linear systemof dimension at least 2, then the Kodaira embedding of L−1 ⊂ H−1 gives a holomorphiccurve f : M → P(H∗). To obtain a Bianchi permutability theorem and a geometricinterpretation of the Backlund transform we define, in contrast to [LP05], the Backlundtransform as the dual curve of f .

Definition 2.15. Let f : M → HPn be a holomorphic curve and α ∈ V ∗ such thatV = L ⊕ kerα. If H ⊂ H0(L) is a basepoint free linear system of dimension at least 2such that f has a Frenet flag, then the dual curve

Bα,H(f) = f † : M → P(H) ,

of f is called the forward Backlund transform of f with parameters α and H. If H =H0(L) is the complete linear system, we call Bα(f) the forward Backlund transform of fwith parameter α.

We will discuss the inverse transformation, the backward Backlund transformation, inSection 2.2.5. In the following, the adjective forward will be dropped until we return tothe topic of the backward transformation.

Remark 2.16. The assumption that f has a Frenet flag is of rather technical nature. Forexample, if

H =< ψ,−ψg >⊂ H0(L)

is a 2–dimensional linear system then f : M → HP1 is given by

L =(−g1

)H

and its dual curve is

L = L† =(g1

)H .

Since L = L−1 is a holomorphic curve in H∗, we see that

∗δ = J δ ,

where J is the induced complex structure on V ∗/L = L−1. Thus, f : M → HP1 is aconformal map. We call f a (generalized) Backlund transform. The requirement that L


has a Frenet flag turns f into a holomorphic curve, i.e., there exists a complex structureJ on L such that

∗δ = δJ .

More generally, if dimH ≥ 2 then we can define the dual curve of f away from theWeierstraß points. Since the Frenet flag extends at least continuously across the Weierstraßpoints, we obtain a continuous map f : M → P (H∗) which is a holomorphic curve awayfrom the Weierstraß points.

To estimate the dimension h0(L) of the space of holomorphic sections of L, in the casewhen M is compact of genus g, recall the Riemman-Roch Theorem (1.53)

h0(L)− h0(KL−1) = degL− g + 1 . (2.21)

From (1.55) we know that the holomorphic structure on KL−1 is given by the exterior

derivative d∇L−1

with respect to the dual connection ∇L−1on L−1. Since α|L ∈ H0(L−1)

is parallel with respect to ∇L−1the linear map

H0(L−1)→ H0(KL−1) : β 7→ ∇L−1β

has kernel spanned by α|L and we obtain

h0(KL−1) = h0(L−1)− 1 ≥ n .

Applying the Riemann–Roch relation (1.53) we get [LP05] that the dimension of the spaceof holomorphic sections of L is given by

h0(L) = h0(L−1) + degL− g .

In particular, if the degree of the line bundle L = f∗T satisfies

degL ≥ 1 + g − n ,

then the complete linear system H0(L) is at least 2–dimensional.

2.2.2 Construction of Backlund transforms from Abelian integrals

We now will explain how one can use Abelian integrals to construct linear systems H ⊂H0(L) see [LP05], [BP]. The holomorphic structure on L is induced by the splittingV = L ⊕ kerα and is given (1.50), (1.51) by the (0, 1)–part of the flat connection ∇Lon L. On the other hand, the bundle KL has a canonical holomorphic structure entirelydetermined by the holomorphic curve f : M → HPn expressed by (1.55). Therefore, theintegrals of holomorphic sections ω ∈ H0(KL) give holomorphic sections ϕ ∈ H0(L) — atleast on the universal cover pr : M →M . Since the section ψ ∈ H0(L) with < α,ψ >= 1has no zeros any linear system H ⊂ H0(pr∗ L) containing ψ is basepoint free.


Given a linear system HKL ⊂ H0(KL) the linear system H ⊂ H0(pr∗ L) obtained byintegrating sections in HKL and including the section ψ with < α,ψ >= 1 is called thelinear system obtained by integration of HKL. Since we have included ψ ∈ H0(L), whichappears as the constant of integration, in the linear system H this procedure is well–defined. Moreover, because ψ ∈ H has no zeros, the linear system H is basepoint free. Tocalculate the dimension of H in case M is compact and HKL = H0(KL) is the completelinear system, we use the Riemann–Roch theorem (1.53) applied to L−1:

h0(KL) = h0(L−1) + degL+ g − 1

and thereforedim H = 1 + h0(KL) = h0(L−1) + degL+ g .

Lemma 2.17 ([LP05]). Let f : M → HPn be a holomorphic curve. Then, for any choiceof hyperplane α ∈ V ∗ not intersecting f , the linear system H ⊂ H0(pr∗ L) obtained byintegration of H0(KL) is basepoint free and has dimension

dim H = h0(L−1) + degL+ g .

Assuming that m + 1 = dim H ≥ 2, we obtain a Backlund transform on the universalcover M of M .

If we only are concerned about local surface theory then spaces of holomorphic sections areinfinite dimensional, Abelian integrals have no periods, and we always obtain Backlundtransforms by integrating sections in H0(KL).

In the case of compact surfaces, genus 0 is exceptional since there are no periods to close.Moreover, for a holomorphic sphere f : S2 → HPn of

degL ≥ −n+ 1

the previous Lemma implies that m + 1 = dim H ≥ 2 since h0(L−1) ≥ n + 1. Thuswe always have Backlund transforms f : S2 → HPm. In [Pet04], [BP] the Backlundtransformation is used to construct soliton spheres.

For surfaces f : M → HPn with genus of M g ≥ 1, we have to close the periods of theBacklund transform. In case, of a torus this can be done if we allow discrete points on Mwhere the conformality of f fails, see [LP05].

Figure 2.6: 1–step Backlund transform of a Willmore sphere in S4, [Pet04],[Hel02]

For simplicity of notation we denote by Bα,ω : M → HP1 the (generalized) Backlundtransform of f : M → HPn with respect to α and the linear system HL ⊂ H0(L) obtainedby integration of HKL = Spanω.


Definition 2.18. Let f : M → HPn be a holomorphic curve and let

f = Bα,ω(f) : M → HP1

be the Backlund transform with respect to the nowhere vanishing ω ∈ H0(KL). Let L =f∗T and H = H∗L ⊂ H0(L−1) the corresponding basepoint free linear system given by theKodaira correspondence.

The Kodaira embedding of L into any (k + 1)–dimensional linear system

H ⊂ H0(L−1) with H ⊂ H , k ≥ 1 ,

is called a 1–step Backlund transform of f . In particular, Bα,ω(f) is a 1–step Backlundtransform of f .

2.2.3 Envelopes and Osculates

In this section we will give a geometric interpretation of the Backlund transformations.Let us first recall the construction of the tangent curves of a holomorphic curve in CPn:The successive higher derivatives of a holomorphic curve in CPn form a holomorphic flag,the Frenet flag. The intersection of the kth osculating flag with a complementary CPn−kgives [GH94, Ch. 2.4] a new holomorphic curve in CPn−k. The analogous construction fora holomorphic curve in HPn also requires the existence of a smooth osculating flag.

Definition 2.19. Given a holomorphic curve f : M → HPn with Frenet flag, we geta holomorphic curve f+ : M → HPn−1, the tangent curve or (first) osculate of f , byintersecting the first flag space V1 of the Frenet flag of f with a HPn−1 ⊂ HPn.

Conversely, f− : M → HPn+1 is called an envelope of f : M → HPn if f is a tangentcurve of f−, i.e.,

(f−)+ = f .

Remark 2.20. The tangent construction preserves holomorphic curves with Frenet flag,and Frenet curves [LP03].

Figure 2.7: Osculate of a Willmore torus in HP2, [Hel02]

For a Frenet curve f : M → HPn with corresponding line bundle L ⊂ V any choice ofa nowhere vanishing holomorphic section ω ∈ H0(KL) gives an envelope: if ϕ ∈ Γ(V )satisfies ∇ϕ = ω then ψ− = ϕ ⊕ 1 is a nowhere vanishing section of the trivial Hn+1–bundle V− = pr∗(V ) ⊕ H over the universal cover M of M , and defines the quaternionic


line bundle L− = ψ−H ⊂ V−. For α− ∈ V ∗− nowhere vanishing and kerα− = V we seethat

L = kerα− ∩ (L− ⊕ im δ−) = kerα− ∩ V1 ⊂ V , (2.22)

since δ−ψ− = ω ∈ H0(KL). Moreover, ω defines N ∈ im H by ∗ω = ωN and gives acomplex structure J on L− by quaternionic linear extension of

Jψ− = ψ−N .

In particular, L− is a holomorphic curve

f− : M → HPn+1 ,

and due to (2.22) an envelope of f . Moreover, f− has a smooth Frenet flag with (V−)0 = L−and

(V−)k = L− ⊕ Vk−1 for k > 0 .

Lemma 2.21 ([LP03]). Let f : M → HPn be a holomorphic curve with Frenet flag andlet ω ∈ H0(KL) be nowhere vanishing. Then the envelope f− given by ω is a holomorphiccurve with Frenet flag. If f is a Frenet curve so is f−.

Proof. It remains to show that f− is Frenet if f is Frenet. Since ω = δ−ψ− is nowherevanishing we can smoothly define B ∈ Hom(V,L−) by

δ−B = 2 ∗A

where A is the Hopf field of the canonical complex structure S of f . Define the complexstructure S− on V− = L− ⊕ V by

S− =(J B0 S

)then S− is the canonical complex structure of f− by (1.35).

Definition 2.22. The kth osculate fk : M → HPn−k is inductively defined to be the firstosculate

fk = (fk−1)+

of the k−1st osculate fk−1 : M → HPn−k+1. Similarly, the kth envelope f−k : M → HPn+k

is defined inductively.

In other words, if the linear system of f is given by H = α, α1, . . . , αn−1, β with αk|Lk 6=0 then the kth osculate is given by

Lk = Vk ∩ kerαk−1 ,

where L ⊂ V1 ⊂ . . . ⊂ Vn is the Frenet flag of f .


ForH = α, . . . , αn−1

incl→ H ⊂ H0(L−1)

denote by π = incl∗ : V → V the induced projection, where V = H∗ and V = H∗. Wecan identify V = Vn−1 via the splitting Vn−1 ⊕ Ln = V . The trivial connection d on V isgiven by the trivial connection d on V via the splitting, i.e.,

d =(

d 0δn−1 ∇

).

Since α ∈ H, the linear system H is basepoint free. If we denote by L the Kodairaembedding of L ⊂ V , then the flag spaces of f are given by

Vj = Vj , j ≤ n− 1,

where we again use the splitting Vn−1 ⊕ Ln = V to identify π(Vj) with Vj . In particular,the osculates of f are given by Lj = π(Lj) = Lj , so that the diagram

f

π

// f1

π

// . . . // fn−1

π

// fn

f // f1// . . . // fn−1

commutes. Moreover, if f : M → HPn is a Frenet curve with canonical complex structuregiven in the splitting V = Vn−1 ⊕ Ln by

S =(S B

0 J

)then f is a Frenet curve with canonical complex structure S: we compute the Hopf fieldA of S in this splitting

A =(A 1

2(∗∇B)′ + 14(BδB +B∇J)

0 0

).

Since imA ⊂ L, we see that im A ⊂ L = L, and S is the canonical complex structure off . We summarize:

Lemma 2.23. Let f : M → HPn be a holomorphic curve with Frenet flag. Then thelinear system H is basepoint free. Moreover, the Kodaira embedding of L into V = (H)∗

gives a holomorphic curve f : M → HPn−1 with Frenet flag

Vj = πVj , j ≤ n− 1 ,

and jth–osculateLj = π(Lj) .

Moreover, if f : M → HPn is a Frenet curve so is the projection f : M → HPn−1.


The Backlund transformation is given in by the two transformations above: a Backlundtransform is a projection of an envelope of f .

Theorem 2.24 ([LP05]). Let f : M → HPn be a holomorphic curve with Frenet flag andlet ω ∈ H0(KL) be a holomorphic 1–form without zeros. Moreover, let α ∈ (Hn+1)∗ be ahyperplane at infinity not intersecting f .

Then the envelope f− : M → HPn+1 of f given by

L− = ψ−H ⊂ Hn+2

with ∇ψ− = ω projects onto the 1–step Backlund transform Bα,ω(f). In particular, the1–step Backlund transform Bα,ω(f) is a holomorphic curve with Frenet flag. If f is aFrenet curve, so is Bα,ω(f).

Proof. Let ψ ∈ H0(L) be the nowhere vanishing holomorphic section with < α,ψ >= 1.By Lemma 1.36 we can write

ω = ψdg

with g : M → H and

H = Spanψ,−ψg ⊂ H0(pr∗ L)

is the linear system obtained by integration of HKL = Spanω. The Backlund transformBα,ω(f) is therefore given by

Bα,ω(f) = [g, 1] .

On the other hand, the Kodaira embedding of L− ⊂ H∗ with respect to the linear systemH = Spanα−, α ⊂ H− ⊂ H0(L−1

− ), is also given by [g, 1] since

d < α, ψ− >=< α,∇ψ− >=< α,ω >= dg

and < α−, ψ− >= 1. The remaining statements follow from Lemma 2.21 and Lemma2.23.

Corollary 2.25. Let f : M → HPn a holomorphic curve with Frenet flag and ω ∈H0(KL) without zeros. Let f− : M → HPn+1 be the envelope of f with respect to ω, andf = Bα,ω(f) the Backlund transform of f with respect to α and ω.

Any 1–step Backlund transform f of f with linear system H ⊂ H0(L−) with H ⊂ H ⊂ H−is a projection of the envelope f− and projects onto the Backlund transform Bα,ω(f). Inparticular, such a 1–step Backlund transform is a holomorphic curve with Frenet flag. Iff is a Frenet curve, then the 1–step Backlund transform is Frenet, too.


f− : M → HPn+1osculate with δ−ψ−=ω //

π

f : M → HPn

1–step BTrrffffffffffffffffffffffffff

BT with parametersα and ω

vvmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm

f : M → HPk

π

Bα,ω(f) : M → HP1

2.2.4 The (n+1)–step Backlund transformation

The 1–step Backlund transform of a holomorphic curve f : M → HPn is defined by Abelianintegrals, in particular it is only given on the universal cover M of M . However, we willshow that after n + 1 successive 1–step Backlund transforms, the resulting holomorphiccurve, the so–called (n+ 1)–step Backlund transform, is globally defined.

Given a holomorphic curve f ji : M → HPn, we denote an osculate of f ji by

f ji+1 : M → HPn−1 ,

a 1–step Backlund transform of f ji by

f j+1i : M → HPn ,

and a projection of f ji by

f j+1i+1 : M → HPn−1 .

We omit the index 0, i.e.,

f j := f j0 , fj := f0j .


We first prove the commutativity of the following diagram

f−1

// f

||xxxxxxxxxx// . . . // fn−3

// fn−2

||yyyy

yyyy

//

fn−1

||yyyy

yyyy

f1

// f11

||yyyy

yyyy

yy// . . . // f1

n−2

// f1n−1

||yyyy

yyyy

f21

// f22

// . . . // f2n−3

~~

...

...

. ..

~~

fn−1n−2

// fn−1n−1

||yyyy

yyyy

fnn−1

(where → denotes the osculating construction, ↓ a projection, and a 1–step Backlundtransformation).

Lemma 2.26. The 1–step Backlund transform Bα,ω(f) : M → HP1 of a holomorphiccurve f : M → HPn, n ≥ 2, is given by a n–step Backlund transform of the (n − 1)st–osculate fn−1 of f .

Proof. In the case n = 2, the first osculate with respect to α ∈ (H3)∗ with L⊕ kerα = V ,is a map f1 : M → HP1. Denote by ψ ∈ Γ(L) the section with < α,ψ >= 1.

Choose α1 ∈ (H2)∗ such that L1 ⊕ kerα1 = H2, and let ψ1 ∈ Γ(L1) be the section with< α1, ψ1 >= 1. The derivative of L satisfies

δψ = ψ1d < α1, ψ >,

and therefore ω1 = δψ ∈ H0(KL1), see Lemma 1.36.

f

δψ // f1

ω1=δψ

f11

By Theorem 2.24 the 1–step Backlund transform of f1 with respect to ω1 and α1 is givenby the projection f1

1 of f since f is an envelope of f1 with respect to ω1.


Let now n > 2, and let Lj be the jth–osculate of f and f−1 the envelope of f with respectto ω. Let

α−1, α, α1, . . . , αn−1, β ∈ (Hn+2)

such thatH− = Spanα−1, α, α1, . . . , αn−1, β

is the basepoint free linear system of f−1 and

αj |Lj 6= 0 .

The linear systemH1 = Spanα−1, α, α1, . . . , αn−1

induces a projection π : V−1 → V 1, where V−1 = H∗−1 and V 1 = (H1)∗, so that

f1 = π(f−1)

is a 1–step Backlund transform of f by Corollary 2.25. By Lemma 2.23 f1 has jth–osculatef1j where each f1

j is the projection of fj−1.

Applying the above argument for n = 2, the map f1n−1 is the 1–step Backlund transform

of fn−1 with respect to the holomorphic 1–form

ωn−1 = δn−2ψn−2 .

Proceeding inductively, fkn−1 is a Backlund transform of fk−1n−1 and, by construction, also

the 1–step Backlund transform of fn−k with parameters αk and ωk = δkψk.

Theorem 2.27. Let f : M → HPn be a holomorphic curve with Frenet flag, and letω ∈ Ω1(End(V )) be a nowhere vanishing 1-form with dω = 0, imω = L, and ∗ω = Jω.Moreover, assume that L = (kerω)⊥ is a holomorphic curve with Frenet flag in HPn.Then the dual curve f : M → HPn of f is a (n + 1)–step Backlund transform of f . Inparticular, the (n+ 1)–step Backlund transform is globally defined.

Remark 2.28. The assumptions of the theorem can be relaxed. If kerω does not containa constant subbundle then L is a holomorphic cure in HPn see Example 1.18, and similararguments as below can be used away from the Weierstraß points of f to show that the(n + 1)–step Backlund transform is defined on M without isolated points. Moreover,the assumption that f is a full curve guarantees that the successive 1–step Backlundtransforms are not constant maps, and can be dropped if we allow more general Backlundtransforms.

Proof. Choose a basis a1, . . . , an+1 of Hn+1 and let α1, . . . , αn denote the dual basis. Since

dωi = (dω)ai = 0 ,


we can define holomorphic 1–forms ω0i ∈ H0(KL) by

ω0i := ωai .

Away from the zeros M0 of ω01 we can define the 1–step Backlund transform fn+1

n :M \M0 → HP1 of f with respect to ω0

1 and α1, i.e.,

fn+1n = Bα1,ω0

1(f) .

We will show that the dual curve L of L = (kerω)⊥ projects onto an n–step Backlundtransform of fn+1

n away from the Weierstraß points of f . Moreover, we will see that f isa n− 1st envelope of a 2–step Backlund transform of f . On the other hand, the followingdiagram is commutative as we have seen before where fn+1

n−1 : M → HP1 is a 2–stepBacklund transform of f , and fn+1 : M → HPn and f2n

n−1 : M → HP1 are (n + 1)–stepBacklund transforms.

f

~~||||

||||

f1

||zzzz

zzzz

zz

f2

~~~~

~~~~

~

...

. ..

~~

...

fnn−1

fn+1

// · · · // fn+1n−1

~~

...

. ..

~~

f2nn−1

In other words, we will show that f = fn+1.

Define g1j : M → H away from the isolated zeros M0 of ω0

1 by

ω0j+1 = ω0

1g1j , j = 1, . . . , n . (2.23)

This way, we obtain on M0 = M \M0

kerω = Spana1g1j − aj+1


and the nowhere vanishing section

ψ =n−1∑k=1

αk+1g1k + α1

spans L = (kerω)⊥ over M0. In affine coordinates, this reads as

L =

g1n...g1

1

1

H .

Since f is a holomorphic curve with Frenet flag, we can compute the kth–osculates fk off given by Vk ∩ ker ak. Note that dg1

j 6≡ 0 since f is a full curve in HPn. In particular,dg1

1 has isolated zeros M1, and away from M1 we can define g2j , j = 1, . . . , n− 1 by

dg1j+1 = dg1

1g2j .

Proceeding inductively we define on M = M \⋃n−1j=0 Mj

dgkj+1 = dgk1gk+1j+ (2.24)

so that the kth osculate is given in affine coordinates by

Lk =

gk+1n−k...

gk+11

1

H .

Note that (2.24) implies thatdgk−1

1 ∧ dgkj = 0 ,

so that[gk1 , 1] : M0 → HP1

is a 1–step Backlund transform of [gk−11 , 1].

Moreover, since ω0j+1 is closed, (2.23) shows that g1

1 is a 1–step Backlund transform offn+1n , and the map

fn+kn−1 = [gk1 , 1]

is a (k + 1)–step Backlund transform of f .

Since (Vn−1)⊥ ⊂ L⊥n−1 the dual curve f of f is given in affine coordinates byf1

...

fn1


with fn = −gn1 . In other words, f projects onto a (n+1)–step Backlund transform of f . Alengthy but straightforward computation shows that the coordinates of f are recursivelygiven by

fk = −gkn−k+1 −n−k∑l=1

gkl fl+k

and that the n− 1st osculate of f satisfies

Ln−1 =(−g1

1

1

)H .

Corollary 1.18 now shows that f is a (n+1)–step Backlund transform of f since f projectsonto a (n+ 1)–step Backlund transform of f and envelopes a 2–step Backlund transform.

Remark 2.29. The previous theorem can be interpreted as Bianchi permutability: theproof shows that if we prescribe n+ 1 holomorphic ω0

i ∈ H0(KL) then there is a commonn–step Backlund transform of the 1–step Backlund transforms Bα,ω0

i(f) of f . Moreover,

this common n–step Backlund transform is, up to Mobius equivalence, algebraically givenby kerω where ω ∈ Ω1(End(V )) is defined by ωai = ωi after a choice of a basis ai of Hn+1.

2.2.5 The backward Backlund transformation

The backward Backlund transformation is the inverse transformation of the forward Backlundtransformation.

Theorem 2.30 (and Definition). If f : M → HPn is a holomorphic curve with Frenetflag and ω ∈ Ω1(End(V )) a nowhere vanishing closed 1–form with

ker ω = Vn−1 ,

and ∗ω = ωS for some adapted complex structure S of f such that im ω is not containedin a proper constant subbundle of V , then

L = im ω

defines a holomorphic curve f : M → HPn, the so–called (n+1)–step backward Backlundtransform of f .

Proof. Let f † : M → HPn be the dual curve of f . Example 1.17 shows that f † is aholomorphic curve with Frenet flag L†k = V ⊥n−k−1 and complex structure J†k = J∗n−k onVk/Vk−1. If we define

ω = ω∗ ∈ Ω1(V ∗) ,


then ω is a nowhere vanishing closed 1–form with

imω = (kerω)† = L† .

Moreover, ω ∈ Γ(KL†) since S∗|L† = Jn is the complex structure of the dual curve on L†,and

∗ω = S∗ω .

Thus, the (n + 1)–step forward Backlund transformation of f † is the holomorphic curvegiven by

(L†) = ((kerω)⊥)† .

In particular,L = (im ω) = (kerω)⊥ = (L†)†

is a holomorphic curve with Frenet flag.

Corollary 2.31. Let f : M → HPn be a holomorphic curve with Frenet flag, and f : M →HPn be the (n+ 1)–step forward Backlund transform of f given by ω ∈ Γ(K Hom(V,L)).

Then f is a (n+ 1)–step backward Backlund transform of f , i.e.,

f = f .

Conversely, given a backward Backlund transform f : M → HPn of f with respect toω ∈ Ω1(End(V )), then f is a (n+ 1)–step forward Backlund transform of f , i.e.,

˜f = f .

Proof. The 1–form ω haskerω = Vn−1

where Vk are the flag spaces of the forward Backlund transform f . Since ω is closed andnowhere vanishing and satisfies ∗ω = ωS for some adapted complex structure S of f , thisyields that

L = imω

is a backward Backlund transform of f .

The 1–step backward Backlund transform of a holomorphic curve f : M → HPn withFrenet flag is given by a closed 1–form ω ∈ Ω1(L) with ∗ω = ωJ : the dual form

ω = ω∗ ∈ H0(KL−1)

is a holomorphic section in KL−1. If β ∈ Γ(pr∗ L−1) satisfies dϕ = ω then β is a holomor-phic section in pr∗(L−1). Assume that β 6∈ H then H and β span a (n+ 2)–dimensionalbasepoint free linear system H and the Kodaira embedding of L into H∗ is a holomorphic


curve f . Its derivative δ, gives after the choice of a hyperplane α ∈ H at infinity notintersecting f , a line bundle

L = V1 ∩ kerα ,

the 1–step backward Backlund transform of f . Here V1 is the first flag space of f .

Note that the 1–step forward Backlund transform of f given by ω = δψ is f , whereψ ∈ Γ(L) is given by < α,ψ >= 1.

Remark 2.32. In the following, we will only prove statements for the forward (or backward)transform: as the proof of Theorem 2.30 shows, we can switch from one transformationto the other by considering the dual curve. The translation of properties of the forwardBacklund transform to the corresponding properties of the backward transform is alwaysgiven by straight forward arguments.

Chapter 3

Applications to Willmore curves

An immersion f : M → R4 of a compact Riemann surface into the 4–space is calledWillmore surface if f is a critical point under compactly supported variations of f of theWillmore functional

W (f) =∫M

(|H|2 −K −K⊥)|df |2 ,

where H is the mean curvature vector of f , K the Gaussian curvature and K⊥ the cur-vature of the normal bundle of f all computed with respect to the induced metric on M .Willmore surfaces have a long history attached to them: [Bla29], [Wil68], [Wei78], [LY82],[Bry84], [Eji88], [Sim93], [Mon00]. For an introduction to the subject see also [Wil93].

More general, if f : M → S4 = HP1 is a holomorphic curve with dimH0(L−1) = n+1 ≥ 2,we can ask under which conditions f is a critical point of the Willmore energy W(f) =2∫M < A ∧ ∗A > under variations ft : M → HP1 of f which preserve the dimension

dimH0(L−1t ) ≥ n + 1 of the space of holomorphic sections. For n = 1 we clearly obtain

the classical Willmore surfaces f : M → S4. However, in general these critical pointsare not necessary Willmore surfaces in S4 if n > 1: since we have a constraint on thevariations, namely to preserve the dimension of the space of holomorphic sections, weallow fewer variations and thus a larger class of examples. Examples for such conformalmaps are some soliton spheres, [Pet04].

On the other hand, using the Kodaira correspondence we can consider L = f∗T embeddedin (H0(L−1))∗ and obtain a holomorphic curve in HPn. The constraint on the variationgives a variation of f : M → HPn by holomorphic curves ft : M → HPn. Since theWillmore energy is given by the holomorphic structure on the holomorphic line bundleL−1, and thus independent of the linear system, f is a critical point under the variationft. In other words, we obtain a Willmore curve in HPn, that is a critical point of theWillmore energy under variations by holomorphic curves in HPn. Willmore curves in HPnbehave very much like Willmore surfaces in S4. In particular, if f : M → HPn is Frenet,then the canonical complex structure is harmonic. Moreover, Willmore spheres in HPn

81

82 CHAPTER 3. APPLICATIONS TO WILLMORE CURVES

have integer Willmore energy [Les], and are given by complex holomorphic data — thus,also any projection f : M → S4 of a Willmore curve in HPn has integer Willmore energyand is given by complex holomorphic data.

We will study the transformations discussed in Chapter 2 in the special case of a Willmorecurve. The Darboux transformation gives a spectral curve of a Willmore torus f : T 2 →S4 with trivial normal bundle, and the Backlund transformation will allow to give aclassification of Willmore spheres f : S2 → S4 in terms of holomorphic data. Thisclassification extends to Willmore tori f : T 2 → S4 with non–trivial normal bundle.The results for Willmore spheres and Willmore tori with non–trivial normal bundle havegeneralization to the case of Willmore spheres f : S2 → HPn and f : T 2 → HPn in HPn.

3.1 Willmore curves in HPn

From now on, M will always denote a compact Riemann surface.

Definition 3.1 (see [LP03]). A holomorphic curve f : M → HPn is called Willmore iff is a critical point of the Willmore energy under compactly supported variations of f byholomorphic curves where we allow the conformal structure on M to vary.

Figure 3.1: Willmore sphere, [Hel02]

In case of a Frenet curve f : M → HPn the Willmore condition can be expressed in termsof harmonicity.

Definition 3.2. The energy functional of S : M → Z := S ∈ End(V ) | S2 = −I isgiven by

E(S) =12

∫M< dS ∧ ∗dS >= 2

∫M< Q ∧ ∗Q > + < A ∧ ∗A > . (3.1)

A map S : M → Z is called harmonic if it is a critical point of the energy functional.

Let d = ∂+ ∂+Q+A the decomposition of the trivial connection d on V with respect toa complex structure S. By changing the complex structure to −S on the domain we getK End−(V ) = K Hom+(V , V ) and ∂ and ∂ on V induce by (1.48) a complex holomorphicstructure on K End−(V ). If we change the complex structure on K End−(V ) to −S then∂ and ∂ give a complex holomorphic structure ∂ on K End−(V ) = K End−(V ), i.e., anantiholomorphic structure ∂ on K End−(V ).

As in [BFL+02, Prop. 5] one shows

3.1. WILLMORE CURVES 83

Theorem 3.3. Let S : M → Z. Then following are equivalent

1. S is harmonic.

2. ∗Q is closed which due to (1.34) is the same as ∗A is closed.

3. Q is antiholomorphic, i.e., ∂Q = 0.

4. A is holomorphic, i.e., ∂A = 0.

Moreover, if f : M → HPn is a Frenet curve and S : M → Z its canonical complexstructure, then S is conformal, i.e.,

< ∗dS, ∗dS >=< dS, dS > .

Since ∂ + ∂ is a complex connection on V with respect to the complex structure S, thedegree of V is given by (1.46)

2π deg(V, S) =∫M< A ∧ ∗A > − < Q ∧ ∗Q > . (3.2)

If f : M → HPn is a Frenet curve with canonical complex structure S then the Willmoreenergy is given in terms of the Hopf field A of S by W(f) = 2

∫M < A ∧ ∗A > so that we

have

Corollary 3.4. Let f : M → HPn be a Frenet curve with canonical complex structure S.Then

E(S) + 4π deg(V, S) = 2W(f) .

Similar techniques as used in the S4–case [BFL+02, Thm. 3] give the classical relationbetween the Willmore condition and harmonicity.

Theorem 3.5 (see [LP03]). A Frenet curve f : M → HPn is Willmore if and only if thecanonical complex structure of f is harmonic, i.e.,

d ∗A = 0 .

Corollary 3.6. Let f : M → HPn be a Frenet curve. Then

(d ∗A)|Vn−1 = 0 .

In particular, if e ∈ Hn+1 such that V = Vn−1 ⊕ eH then f is Willmore if and only if

(d ∗A)e = 0 .


Proof. For ψ ∈ Γ(Vn−1) we have

(d ∗A)ψn−1 = (d ∗Q)ψn−1 = d(∗Qψn−1)−Q ∧ dψn−1 = − ∗Q ∧ δn−1ψn−1 ,

where we used (1.34) and that Q|Vn−1 = 0 for the Hopf field of the canonical complexstructure (1.33). But Q ∧ δn−1 = 0 by type.

Example 3.7. As we have seen in Example 1.34, holomorphic curves f : M → HPn withzero Willmore energy are exactly the twistor projections of holomorphic curves in CP2n+1.These provide the simplest examples of Willmore curves. Note that even in this case, theWillmore condition does not guarantee the smooth existence of the canonical complexstructure S of the holomorphic curve in the Weierstraß points. However, a regularityresult of Helein [Hel04] on harmonic maps, can be used to show that a Willmore curve fis Frenet if S can be extended continuously across the Weierstraß points [LP03].

Example 3.8. The notion of a Willmore curve carries over to the dual curve of a Frenetcurve f : M → HPn: the Hopf field Q† of f † satisfies (1.32)

d∗ ∗Q† = −(d ∗A)∗

so that f is Willmore if and only if f † is Willmore.

We have the Kodaira correspondence between holomorphic curves f : M → HPn andbase point free linear systems H ⊂ H0(L−1). For a Willmore curve f : M → HPn, it isnatural to ask for which choices of basepoint free linear systems H ⊂ H0(L−1) the inducedholomorphic curve L ⊂ H−1 is again Willmore.

Proposition 3.9. Let f : M → HPn be a Willmore curve. Let L ⊂ V and H ⊂ H0(L−1)be the corresponding line bundle and basepoint free linear system. Let H ⊂ H0(L−1) be alinear system with H = V −1 ⊂ H so that the map f : M → HPm given by the Kodairacorrespondence has a canonical complex structure which extends continuously across theWeierstraß points. Then f a Willmore curve in HPm where m = dim H.

Proof. Let ft : M → HPm be a variation of f so that the compact support K does notcontain Weierstraß points. Without loss of generality, we can assume that ft is unramifiedon K. Then π : V = H−1 → V defines a variation of f by Frenet curves ft : M → HPnby π(Lt) = Lt. Since the Willmore energy only depends on the holomorphic structure onL−1 and not on the linear system, we see that

∂

∂tW(ft) =

∂

∂tW(ft) = 0 .

The usual arguments, see [LP03], show that the canonical complex structure of f is har-monic on K, i.e.,

d ∗ A = 0

away from the Weierstraß points. With [Hel04] the canonical complex structure extendssmoothly into the Weierstraß points, and therefore f is a Frenet curve.

3.1. WILLMORE CURVES 85

In general, projections of Willmore curves L ⊂ V into flat subbundles V ⊂ V fail to beWillmore, see [Pet04].

Proposition 3.10. Let f : M → HPn be a Willmore curve with canonical complexstructure S and let H ⊂ H0(L−1) be the corresponding linear system. Let H be an Sstable basepoint free linear system H ⊂ H ⊂ H0(L−1) with m = dim H ≥ 2. Then

L = π(L) ⊂ V = H−1

defines a Willmore curve f : M → HPm. Here π : V = H−1 → V is the canonicalprojection.

Proof. Since f is a holomorphic curve, the line bundle L is full, i.e., L is not contained ina lower dimensional flat subbundle of V . The kernel H⊥ = kerπ of π is d stable whichshows that π|L 6= 0. Since H is a linear system the induced connection d on V satisfiesπd = dπ. Moreover,

πS =: Sπ

defines a complex structure on V since kerπ = H⊥ is S stable. The complex holomorphicstructures d′′+ and d′′+ on V and V given by the complex structures S and S are related by

d′′+π = πd′′+ .

Since d′′+ and d′′+ stabilize L and πL respectively, the map π|L is a complex holomorphicmap. In particular, the zeros of π|L are isolated and the complex bundle imπ|L can beextended smoothly across the zeros. In other words, imπ|L defines a complex quaternionicline bundle L. Note that Lp = πLp away from the isolated zeros of π|L.

Let L ⊂ V1 ⊂ . . . ⊂ V be the Frenet flag of f . Since ππL = πLπ we see

δ0π|L = πδ0 .

If δ0 = 0 then V1 is contained in the flat bundle L + kerπ which has rank ≤ n sincedim kerπ = rankV − rank V ≤ n − 1. This contradicts the assumption that L is a fullcurve in V , i.e., the assumption that δk 6= 0 for k = 0, . . . , n− 1. Thus the map δ0 6= 0 iscomplex holomorphic since

∗δ0 = Sδ0 = δ0S ,

and defines a vector bundle V1. Clearly, V1 extends πV1.

Proceeding inductively, we see that δkπ|Vk = πδk and δk 6= 0 for all 0 ≤ k ≤ rank V −2. Inparticular, L is a full curve in V with Frenet flag Vk = πVk. Moreover, ∗δk = Sδk = δkSyields that S is an adapted complex structure.

By construction A = 12 ∗(dS)′ and A = 1

2 ∗(dS)′ satisfy Aπ = πA, hence S is the canonicalcomplex structure of f . In particular f is a Frenet curve, and

d ∗ Aπ = πd ∗A = 0 .

shows that f is Willmore.


Remark 3.11. If dim H = 1 the same arguments as in the proof above show that (π(L), πS, πd)defines a flat complex quaternionic line bundle.

Since the Hopf fields A and Q of a Willmore curve are holomorphic, the zeros of A and Qare isolated. Since A|L = 0 at p ∈M implies [Les] that A ≡ 0 in a neighborhood of p, weobtain:

Corollary 3.12. Let S be the canonical complex structure of a Willmore curve f : M →HPn.

1. If A 6= 0 then the set

M := p ∈M | Lp ⊂ kerAp

has no inner points.

2. If Q 6= 0 then the set

M := p ∈M | imQp ⊂ (Vn−1)p

has no inner points.

Remark 3.13. A flat connection ∇ on a complex quaternionic vector bundle (W,S) iscalled Willmore connection [FLPP01, Sec. 6.1] if S is harmonic, i.e., d∇ ∗A = 0.

Examples include the trivial connection on (V, S) where S is the canonical complex struc-ture of a Willmore curve, and the family of flat connections (2.12) associated with aconstant mean curvature surface where W = V/L has rank 1, and S = J is the complexstructure given by the harmonic Gauss map of f .

In general, even if W = V has rank 2, the harmonic complex structure S will not be thecanonical complex structure of a Frenet curve. But if rankA = 1 then ∂A = 0 implies thatthe image of A defines a ∂ holomorphic line bundle. Thus, L = imA ⊂ V is a Willmorecurve if S is the canonical complex structure of L, i.e., if the Hopf field Q|Vn−1 = 0 vanisheson the flag space Vn−1 of L.

Moreover, if L is Willmore then the connections ∇λ = ∇ + (λ − 1)A are flat for allλ = α + βS, α, β ∈ R, α2 + β2 = 1. Denote by Lλ the line bundle L considered assubbundle in (V,∇λ). If we decompose ∇λ with respect to the complex structure S ofL then Qλ = Q. Thus S is the canonical complex structure of Lλ and we obtain theassociated family of Willmore curves Lλ. Its Willmore energy is given byW(Lλ) =W(L).

Notice, that though ∇ is trivial, the Willmore curves of this family may have holonomy.

3.2. THE DARBOUX TRANSFORMATION ON WILLMORE SURFACES IN S4 87

3.2 The Darboux transformation on Willmore surfaces inS4

There are two ways to define a Darboux transformation on Willmore surfaces f : M → S4.One of them comes from the observation that Willmore surfaces behave in many ways likethe rank 2 analog of constant mean curvature surfaces. Solving a Ricatti type equationon a rank 2 bundle we obtain a new Willmore surface. On the other hand, we defined ageneral Darboux transformation on conformal maps into the 4-sphere. We will see howthese two transformations are related.

Let f : M → S4 be a Willmore immersion and let T be a solution of the Ricatti equation

dT = 2ρT ∗QT − 2 ∗A (3.3)

for ρ ∈ R with initial condition

(T − S)2 = ρ−1 − 1 (3.4)

at some point p0 ∈ M . Here A and Q are the Hopf fields of the canonical complexstructure S of f . In [BFL+02, Thm. 12] it is shown that the line bundle

L = T−1L

is a Willmore surface in S4, and (3.4) holds on M . Note that (3.4) is equivalent to

T−2 = ρ(1− ST−1 − T−1S)

so that(2b−1(T + Sρ))2 = 4b−2ρ(1− ρ) = −1

where a2 − b2 = 1, and ρ = 1−a2 . Thus, if we define

S] = 2(T−1 + Sρ)b−1

then S] is a complex structure on V . Moreover, let λ = a+ bS] and define the connectiond] by

d] = d+ 2 ∗AT−1 = d+A(λ− 1) .

Using the Ricatti equation (3.3), we get

d](S]b

2ρ) = d](T−1ρ−1 + S)

= (dT−1)ρ−1 + 2 ∗Q− 2 ∗A+ 2[∗AT−1, T−1ρ−1 + S]= 0 .

If we define the family of I–complex connections

dµ = d+A(µ− 1),


where µ = a+ bI, then the Willmore condition d ∗ A = 0 gives that dµ is a family of flatconnections. Moreover,

dµ|Eµ = d]|Eµwhere Eµ is the +i eigenspace of S]. In particular, d] is a flat connection since V =Eµ ⊕ Eµj. Note that, since d]S] = 0, the ±i eigenspaces Eµ and Eµj of S] are also theeigenspaces of the monodromy of d].

For any parallel section ψ] ∈ Γ(pr∗ V ) of d] the projection ϕ] = πL(ψ]) ∈ H0(pr∗ V/L) isa holomorphic section since

dψ] = −(2 ∗AT−1)ψ] ∈ Ω1(pr∗ L) ,

and thusDϕ] = (πdψ])′′ = 0 .

Moreover, πL(dψ]) = 0 shows that ψ] is the canonical lift of the holomorphic section ϕ]

and ψ]H is a Darboux transform of f .

Lemma 3.14. Every solution of the Ricatti equation (3.3) with initial condition (3.4)gives a Willmore surface f : M → S4 and a Darboux transform f ] : M → S4 of f .

Remark 3.15. In [Sch02], see also[Boh], it is shown that the spectral curve of a Willmorecurve has finite genus. It is still an open problem if the points of the spectral curve of aWillmore surface are again Willmore, and more specifically if the two transforms f and f ]

of a Willmore surface in S4 coincide. The examples of [Ber01], where Darboux transformsof the Clifford torus are constructed which are not constraint Willmore, shows that inthe case of a Willmore torus a general Darboux transform will not even be constraintWillmore. We expect this kind of behavior only at exceptional points of the spectralcurve.

3.3 The Backlund transformation on Willmore curves

Bryant [Bry84] gave a classification result for Willmore spheres f : S2 → R3. This resultcan be generalized to the case of Willmore spheres f : S2 → S4, see [Eji88], [Mon00]:A Willmore sphere in S4 is either the twistor projection of a holomorphic curve in CP3,the dual curve of such a twistor projection, or a minimal sphere in R4 with planar endsafter choosing a suitable point at infinity. A similar result can be obtained [LPP05] forWilllmore tori f : T 2 → S4 with trivial normal bundle by discussing the monodromyof the associated family of flat connections. We present a different approach [LP] usingthe Backlund transformation to construct sequences of Willmore surfaces. This approachextends to the case of Willmore spheres in HPn and shows that every Willmore sphere inHPn has integer Willmore energy and is given by complex holomorphic data [Les]. Herewe also give a generalization of [LP] to Willmore tori in HPn with non–zero degree of thecorresponding bundle L.

3.3. THE BACKLUND TRANSFORMATION ON WILLMORE CURVES 89

3.3.1 Sequences of Willmore curves

Recall that the Willmore condition of a Frenet curve can be expressed in terms of the har-monicity of the canonical complex structure. Recall the ∂, ∂ transformation for harmonicmaps h : M → CPn: the (0, 1)–part δ′′E of the derivative of the line bundle E = h∗Tis holomorphic. The image of δ′′E defines a complex bundle of rank 2, and after orthog-onal projection onto the orthogonal complement E⊥ of E, one gets a line subbunde ofCn+1. The corresponding h : M → CPn is a new harmonic map. This way, one con-structs sequences of harmonic maps which can be used for classification of harmonic mapsh : M → CPn with large (in terms of the genus of M) degree of h, [EW83], [Wol88].

In case of a Willmore curve in HPn with harmonic canonical complex structure S the (1, 0)and (0, 1)–parts of the derivative of S are essentially (1.34) the Hopf fields A and Q. Thekernels and images of A and Q are smooth vector bundles and define holomorphic curvesin HPn since A and Q are holomorphic sections by the harmonicity of S see Theorem 3.3.Of course, the holomorphic curves imA = L and (kerQ)⊥ = L† give no new surfaces.However,

f = ((kerA)⊥)†

andf = imQ

give the (n+1)–step forward and backward Backlund transform of f as defined in Chapter2 for the special choices ω = ∗A and ω = ∗Q. In [Les] it is shown that if f : M → HPn isWillmore and Frenet such that L = ((kerA)⊥)† is a Frenet curve too, then f is Willmoreand the Hopf field Q of the canonical complex structure S of f satisfies

Q = A .

It remains to show that the (n+1)–step Backlund transformation on Willmore curves withω = ∗A is Frenet. We prove the statement first for the 1–step Backlund transformation.

Lemma 3.16. Let f : M → HPn be a Willmore curve with smooth canonical complexstructure S such that A 6≡ 0. Choose b ∈ Hn+1 such that Vn−1⊕bH = V and Vn−1⊕bH = Vwhere Vn−1 and Vn−1 are the flag spaces of L and L respectively. Then the 1–step Backlundtransform f1 : M → HPn given by ω = ∗Ab has a smooth canonical complex structure S1

on the universal cover M of M .

Proof. Let f− : M → HPn+1 be the first osculate of f given by d−ψ− = ω where ω =∗Ab ∈ H0(KL), and let β ∈ Γ(V ∗) with β(b) = 1 and β|Vn−1

= 0. Then

J−ψ− = −ψ < β, Sb >

defines a complex structure on L− and f− is a Frenet curve [LP03] with canonical complexstructure S− given by

S− =(J− B0 S

),


where B ∈ Γ(Hom(V,L)) is defined by B = 2ψβ. Since the 1–step Backlund transformof f with respect to ω is given by a projection of f− Lemma 2.23 shows that the 1–stepBacklund transform is Frenet, too.

To conclude that the (n + 1)–step Backlund transform is Frenet, we have to assure thatthe 1–step Backlund transform of a Willmore curve is Willmore.

Lemma 3.17. Under the assumptions of the previous Lemma, the 1–step Backlund trans-form of f is Willmore.

Proof. We adapt the proof of [BFL+02, Prop. 16] for the 1–step Backlund transform ofa Willmore surface in S4 to the case of Willmore curves in HPn. By the previous lemmathe 1–step Backlund transform f of f is a Frenet curve so that it is enough to show, seeCorollary 3.6, that

(d ∗ A)|eH = 0 ,

where d is the flat connection on V , and A the Hopf field of the canonical complex structureS of f . Moreover, e ∈ Hn+1 such that

Vn−1 ⊕ eH = V .

Note that we can identify Vk = L⊕Vk−1, where Vk are the flag spaces of f . In particular,eH = Ln−1. We decompose S in the splitting L⊕ Vn−1 = V as

S =(J B

0 S

).

We will show thatδ0(2∇B + ∗A)|Ln−1 = ∗Aδn−1 . (3.5)

Let R be the parallel homomorphism R ∈ Γ(Hom(bH, L)) given by

Rb = ψ ,

where < α, ψ >= 1 and α ∈ (Hn+1)∗ is a hyperplane at infinity such that ker α = Vn−1.Since ∗Ab = δ0ψ the equation (3.5) implies that

∗A|Ln−1 = Rδn−1 − 2(∇B)|Ln−1 .

Therefore, since Ln−1 = eH is a constant bundle with respect to the connection d on V ,we have

(d ∗ A)|Ln−1 = d(Rδn−1) + 2(d∇B)|Ln−1 = 0 , (3.6)

where we used that R is parallel with respect to the induced connections and dδn−1 = 0by the flatness of d.


It remains to show (3.5). We use the fact that the complex structure S on Vn−1 given byS via the splitting L ⊕ Vn−1 = V also occurs when decomposing the canonical complexstructure S of f with respect to the splitting Vn−1 ⊕ bH = V :

S =(S B

0 J

).

The conditions (1.35) and (1.33) that S and S are the canonical complex structures of fand f respectively give

2δ0B = S ∗ ∇S − ∇S (3.7)

and2Bδn−1 = S ∗ ∇S + ∇S . (3.8)

Differentiating (3.7) and (3.8) and comparing the results, we obtain

−δ0 ∧∇B = ∇B ∧ δn−1 ,

or, using the identification ω ∧ η = ω ∗ η − ∗ωη of a 2–form with its quadratic form,

− δ0(∗∇B − J∇B) = ((∇B)J − ∗∇B)δn−1 . (3.9)

We denote by η = 4 ∗ A|Vn−1 so that

η = −2(∇B)′ + B ∗ δ0B + B ∗ ∇S .

Since S is a complex structure, i.e. S2 = −1, we have

JB + BS = 0 .

Together with (3.7) we get

J η = −J∇B − ∗∇B − Bδ0B − B∇S ,

so that

Sδ0(2∇B + η) = δ(2J∇B − J η)= ((∇B)J − ∗∇B)δn−1 − δ0(δ0 + ∇S)

= ((∇B)J − ∗∇B)δn−1 −14

(S ∗ ∇S − ∇S)(S ∗ ∇S + ∇S) .(3.10)

On the other hand, if we decompose the trivial connection d in the splitting Vn−1⊕bH = Vas

d =(∇ 0δn−1 ∇b

),

then the Hopf field A of S is given in this splitting as

−4 ∗A =

(−4 ∗AS 2(∇B)′ −B ∗ δn−1B −B ∗ ∇bJ

0 0

),


andδn−1B = 2 ∗AJ

holds. Note that δn−1|Ln−1 = δn−1. As before, S2 = −1 implies SB + BJ = 0, anddifferentiating this equation

−S∇B −B∇bJ = (∇B)J + (∇S)B .

As similar computation as before together with the above equation gives for η = 4 ∗A|bH

Sη = (∇S)B + (∇B)J − ∗∇B −Bδn−1B

so that

Sηδn−1 = ((∇B)J − ∗∇B)δn−1 + (∇S −Bδn−1)Bδn−1

= ((∇B)J − ∗∇B)δn−1 −14

(S ∗ ∇S − ∇S)(S ∗ ∇S + ∇S) .

Comparing to (3.10) this yields (3.5), and we can conclude that the 1–step Backlundtransform f is a Willmore curve since (3.6) and Corollary 3.6 show that

d ∗ A = 0 .

Corollary 3.18. The (n+ 1)–step Backlund transforms f and f of f are Frenet curves.

Combining the Corollary with the result in [Les] we obtain:

Theorem 3.19. The (n + 1)–step forward and backward Backlund transforms f : M →HPn and f : M → HPn of a Willmore curve f : M → HPn are Willmore curves in HPnwith Willmore energies

W(f †) =W(f) and W (f) =W(f †) . (3.11)

3.3.2 Finite sequences

In what follows we only consider (n + 1)–step Backlund transforms as these are globallydefined objects. For simplicity of notation we write “f is a forward Backlund transform”when referring to a (n+ 1)–step transform forward Backlund transform given by kerA.

We now discuss the case when the sequence of successive forward Backlund transforms isfinite, that is that after the kth Backlund transform fk : M → HPn of a Willmore curvef : M → HPn the k + 1st Backlund transform does not exist as a full curve in HPn.


One reason for this to happen is that the Hopf field Ak of fk vanishes identically. Thiscould happen in the first step, in which case f itself has Hopf field A ≡ 0, and f is thetwistor projection of a holomorphic curve in CP2n+1. Otherwise, i.e., for k ≥ 1 we know

that the backward Backlund transform fk = fk−1 : M → HPn is a full curve in HPn

where we denote by f0 = f the original Willmore curve in HPn.

We denote for a Frenet curve f : M → HPn by

δk = δk−1 . . . δ0 ∈ H0(Kk Hom(L, Vk/Vk−1)) , 1 ≤ k ≤ n ,

the complex holomorphic section which is given by the composition of the derivatives ofthe successive Frenet spaces.

Lemma 3.20. Let f : M → HPn be a Willmore curve with canonical complex structureS such that the backward Backlund transform f : M → HPn given by L = imQ is a fullcurve in HPn. If

AδkQ = 0

for all k = 1, . . . , n− 1 then −S is the canonical complex structure of L.

Proof. If AQ ≡ 0 then L = imQ ⊂ kerA so that Q and A stabilize L. [Les] shows thatthis implies

∗δ0 = −Sδ0 = −Sδ0S .

By applying this argument successively for all flag spaces, we see that −S is an adaptedcomplex structure. But the Hopf field A of −S is given by A = Q so that

im A = imQ ⊂ L

and −S is the canonicacl complex structure of f .

A special case when the assumptions of the previous lemma are satisfied is when the Hopffield of f vanishes identically, i.e., A ≡ 0, that is when f is the twistor projection of aholomorphic curve h : M → CP2n+1.

Corollary 3.21. If f : M → HPn is the twistor projection of a holomorphic cure inCP2n+1 and has a backward Backlund transform L = imQ as full curve in HPn then fhas Q ≡ 0, and is the dual curve of the twistor projection of a holomorphic curve inCP2n+1.

Applying this result to the case of the sequence of forward Backlund transforms we get

Corollary 3.22. If the kth forward Backlund transform fk of f is the twistor projectionof a holomorphic curve in CP2n+1 then 0 ≤ k ≤ 1, and f or f † is the twistor projectionof a holomorphic curve in CP2n+1. Moreover,

W(f) ∈ 4πN .


Now, assume that Aj , Qj 6≡ 0 for all 0 ≤ j ≤ k. Then ker Ak does not define a full curvein HPn if and only if ker Ak contains a constant subbundle W ⊂ ker Ak, see Example 1.18.

By dualization, i.e., switching to f † instead of f , we can use the backward sequence insteadof the forward one. Moreover, if L ⊂ W ⊂ V where W is a proper subspace of V withdimW 6= 1 then we consider the sequence of f instead of the sequence of f . Note thatW(f)−W(f) ∈ 4πN so that f has integer Willmore energy if f has W(f) ∈ 4πN.

So, we are left to discuss sequences of Backlund transforms which terminate because thebackward Backlund transform is a constant point in HPn.

Assume that f : M → HPn with n ≥ 2 has constant backward Backlund transformL = W . Then the projection π : V → V/L = V defines a Willmore curve in HPn−1, seeProposition 3.10, with canonical complex structure

Sπ = πS .

In particular, Qπ = πQ = 0 so that f is the dual curve of a twistor projection of aholomorphic curve in CP2n−1. In particular,

W(f) =W(f) ∈ 4πN .

In case f : M → S4 has constant backward Backlund transform f the standard argument[BFL+02, Sec. 11.2] shows that f is a minimal surface in R4 when choosing the point atinfinity as L: all mean curvature spheres of f pass through L.

We summarize:

Theorem 3.23. Let f : M → HPn be a Willmore surface such that the sequence ofBacklund transforms is finite where we allow the Backlund transforms to be full curves inlower dimensional HPk’s. Then f has Willmore energy

W(f) ∈ 4πN .

3.3.3 Willmore spheres and Willmore tori

We show that in the case of Willmore spheres and Willmore tori the sequence of Backlundtransforms is finite.

Let f : M → HPn be a Willmore curve with canonical complex structure S. Since

δi = δi−1 . . . δ0 ∈ H0(Ki Hom+(L, Vi+1/Vi))

is complex holomorphic, we have

ord δn = n degK + deg(V/Vn−1)− degL .


The Hopf fields A and Q are holomorphic sections in the appropriate bundles and so is[Les]

AQ ∈ H0(K2 Hom+(V/Vn−1, L)) .

Therefore, if AQ 6≡ 0 then

0 ≤ ord(AQ) = 2 degK + degL− deg(V/Vn−1) = (n+ 2) degK − ord δn .

In the case of sphere, i.e., degK < 0, this implies AQ ≡ 0 since otherwise

ord(AQ) = (n+ 2) degK − ord δn < 0 .

In the case of a torus, i.e., degK = 0, the above inequality gives for AQ 6≡ 0, that A andQ are nowhere vanishing, and

ord δn = 0

In particular, a Willmore torus f : T 2 → HPn with AQ 6≡ 0 is unramified.

If AQ ≡ 0 and the backward Backlund transform f is not a point in HPn, i.e., δ 6≡ 0, thenit is shown in [Les] that

∗δ = −Sδ = −δS ,

and thusAδQ ∈ H0(K3 Hom+(V/Vn−1, L)) .

As before this implies AδQ ≡ 0 in case of a Willmore sphere. Moreover, for a Willmoretorus with AδQ 6≡ 0 we again see that ord δn = 0. In particular, if we assume thatthe backward Backlund transform of a Willmore sphere f : S2 → HPn is a full curvein HPn, we see by proceeding inductively that A ≡ 0. In the case of a Willmore torusf : T 2 → HPn with f : T 2 → HPn full, we have A ≡ 0 or f : T 2 → HPn is unramified.

Lemma 3.24. If f : S2 → HPn is a Willmore sphere in HPn then the forward andbackward Backlund transforms are not full curves in HPn unless f or f † is the twistorprojection of a holomorphic curve in CP2n+1.

If the Willmore torus f : T 2 → HPn (or its dual f †) is not obtained by a twistor projectionof a holomorphic curve in CP2n+1 and if the backward (or forward) Backlund transformf : T 2 → HPn is a full curve in HPn then f : T 2 → HPn is unramified.

Moreover, in the case of a Willmore sphere f : S2 → HPn with backward Backlundtransform f : M → HPk in HPk, k < n, the above computations show that the canonicalcomplex structure of f is given by −S. This allows to extend the argument used for aconstant backward Backlund transform f to Backlund transforms with L ⊂W ⊂ V sinceW is then S–stable. Again, f projects onto a dual curve of a twistor projection of aholomorphic curve in CP2m+1 for m = n− k:

Theorem 3.25 (see [Les]). A Willmore sphere f : S2 → HPn has integer Willmore energy

W(f) ∈ 4πN .


If n 6= 1 then f comes from the twistor projection of a holomorphic curve h : S2 → CP2k+1

for some k ≤ n. If n = 1 and f does not arise from a twistor projection of a holomorphiccurve h : S2 → CP3 then, after choosing a suitable point at infinity, f is a minimal spherein R4 with planar ends.

For a Willmore curve f : M → HPn the Hopf field A ∈ H0(K Hom+(V /Vn−1, L)) is aholomorphic section where L is the forward Backlund transform of f given by kerA andVi denotes the Frenet flag of L, in particular, we have

0 ≤ ordAδn = deg(Kn+1 Hom+(L, L)) = (n+ 1) degK + degL− deg L .

Now, let f : T 2 → HPn be a Willmore torus, and assume that there exists an infinite seriesof successive Backlund transforms fm. We telescope the above inequality and obtain, usingdegK = 0,

0 ≤ degL− deg Lm (3.12)

The degree of Lm is given by the Plucker formula (1.59) as

deg Lm =1

4π(n+ 1)(W(fm)−W((fm)†)) ,

since the Backlund transforms fm are all unramified, see Lemma 3.24, and thus ord Hm =0. Using (3.11) we have W(fm−1) =W((fm)†) so that

0 ≤ degL− 14π(n+ 1)

(W(fm)−W(fm−1))

Telescoping this inequality again

0 ≤ mdegL− 14π(n+ 1)

(W (fm)−W (f))

we finally get

− 14π(n+ 1)

W (f) ≤ mdegL .

Since we can assume without loss of generality that degL < 0 (otherwise consider the dualWillmore curve L† which has (1.58) in case of an unramified torus degL† = −degL), thisshows that the sequence of a Willmore torus with degL 6= 0 is finite. Theorem 3.23 thusshows

Theorem 3.26. Every unramified Willmore torus f : T 2 → HPn with degL 6= 0 hasinteger Willmore energy

W(f) ∈ 4πN .

Remark 3.27. A Willmore torus f : T 2 → S4 in the 4–sphere is given by complex holo-morphic data: If f has trivial normal bundle then f has a spectral curve of finite genus,and is given by theta functions on the spectral curve [Sch02], [FPPS92].


If f has non–trivial normal bundle then f comes from the twistor projection of a holo-morphic curve h : T 2 → CP3, or is a minimal torus in R4 with planar ends after choosingan appropriate point at infinity.

To obtain a similar result for Willmore tori in HPn, n ≥ 2, with degL 6= 0, one has togain control on the canonical complex structure of a Backlund transform to “reconstruct”the Willmore torus from its Backlund transform f even if f is not a full curve in HPn.

Moreover, we conjecture that Willmore surfaces f : M → HPn with large degree of L (interms of the genus of M) have integer Willmore energy, and are given, at least in casen = 1, as twistor projections of holomorphic curves in complex projective space or areminimal surfaces in R4. To prove such a result, one has to find sharper estimates on thevanishing orders of the involved holomorphic sections A,Q and δn.

Index

Backlund transform(n+ 1)–step backward, 78(n+1)-step, 731-step, 691-step backward, 80forward, 66with parameters α and H, 66with parameter α, 66

basepoint free linear system, 39Bianchi permutability, 51

canonical complex structure, 31complex structure

adapted, 18, 27conformal Gauss map, 19curve

complex holomorphic, 25dual, 29Frenet, 33full, 24holomorphic, 17, 24unramified, 33

Darboux pair, 49Darboux transform, 56

classical, 48singular, 58

degree of a vector bundle, 37derivative, 17dual curve, 29

energy functional, 82envelope, 69

kth, 71

Frenet curve, 33Frenet flag, 26

Fundamental Lemma, 16

harmonic, 83holomorphic

section, 35structure, 35vector bundle, 35

holomorphic structureanti–, 37

Hopf fields, 21hyperplane at infinity, 28

linear system, 35basepoint free, 39

mean curvature, 19mean curvature sphere congruence, 19mean curvature vector, 19

mixed structure, 25

normalright normal, 16left normal, 16

orderof a holomorphic section, 35of a linear system, 35

osculatekth, 71first, 69

Plucker relation, 44point at infinity, 20prolongation, 64

Ricatti equation, 49Riemann–Roch Theorem, 42

98

INDEX 99

spectral curve, 63sphere congruence, 18

envelopes a surface, 18left envelopes surface, 56mean curvature sphere congruence, 19

tangent curve, 69tautological bundle, 16twistor lift, 26twistor projection, 25two–sphere, 17

Weierstraß gap sequence, 35Weierstraß points, 26Willmore

connection, 86curve, 82energy of a Frenet curve, 33energy of a holomorphic bundle, 36energy of a holomorphic curve, 40functional, 22

100 INDEX

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