willmore surfaces of ℝ4 and the whitney spherehera.ugr.es/doi/15003693.pdf · willmore surfaces...

Annals of Global Analysis and Geometry19: 153–175, 2001.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

153

Willmore Surfaces ofR4 and the Whitney Sphere?

ILDEFONSO CASTRO1 and FRANCISCO URBANO21Departamento de Matemáticas Escuela Politécnica Superior, Universidad de Jaén, 23071 Jaén,Spain. e-mail: [email protected] de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain.e-mail: [email protected]

(Received: 20 July 1999; accepted: 6 January 2000)

Abstract. We make a contribution to the study of Willmore surfaces in four-dimensional EuclideanspaceR4 by making use of the identification ofR4 with two-dimensional complex Euclidean spaceC2. We prove that the Whitney sphere is the only Willmore Lagrangian surface of genus zero inR4

and establish some existence and uniqueness results about Willmore Lagrangian tori inR4 ≡ C2.

Mathematics Subject Classifications (2000):53C40, 53C42.

Key words: Lagrangian surfaces, Whitney sphere, Willmore tori.

1. Introduction

Given a compact surface6 in Rn, the Willmore functional is defined by

W(6) =∫6

|H |2 dA,

whereH denotes the mean curvature vector of6 and dA is the canonical measureof the induced metric. An absolute minimum forW is 4π , which is only attained forround spheres. When6 is a torus, Willmore [21] conjectured that the minimum is2π2, and it is attained only by the Clifford torus. Although several partial answershave been obtained (see [10, 13, 17]), the conjecture still remains open. In [19],Simon proved the existence of an embedded torus which minimizes the Willmorefunctional. Given that the Clifford torus is Lagrangian inR4, Minicozzi proved in[11] the existence of an embedded Lagrangian torus which minimizes the Willmorefunctional in the smaller class of Lagrangian tori.

Other authors are interested in the study of critical surfaces of the Willmorefunctional (the so-calledWillmore surfaces), and in the construction of examplesof Willmore surfaces. In [2], Bryant initiated this study, classifying the Willmoresurfaces of genus zero inR3 (or S3, because the Willmore functional as well asthe Willmore surfaces are invariant under conformal transformations of ambientspace). He proved thatany compact Willmore surface of genus zero inR3 is the? Research partially supported by a DGICYT grant No. PB97-0785.

154 ILDEFONSO CASTRO AND FRANCISCO URBANO

compactification by an inversion of a complete minimal surface ofR3 with finitetotal curvature and embedded planar ends.

In [16], Pinkall constructed, using the Hopf fibration5: S3 −→ S2 and elasticcurves ofS2, a wide family of Willmore flat tori inS3. Unlike the Bryant examples,these tori do not come from minimal surfaces ofR3, or minimal surfaces ofS3,or minimal surfaces ofH3. It is also interesting to note that these tori, which arealso Willmore surfaces inR4, are Lagrangian surfaces inR4 with respect to someorthogonal complex structure.

Following Bryant’s ideas, and using different approaches [7, 12, 14], Ejiri,Montiel and Musso classified Willmore compact surfaces of genus zero inR4 (orS4). Besides the Willmore compact surfaces of genus zero coming from completeminimal surfaces ofR4 with finite total curvature and planar ends, a new family ofWillmore spheres has also appeared. These spheres can be described as projectionsunder the Penrose twistor fibrationCP3 −→ S4 of holomorphic curves ofCP3.

In [4], the authors studied the Willmore functional for Lagrangian compactsurfaces6 of genus zero ofR4. As they cannot be embedded, using a result of Liand Yau [10],W(6) ≥ 8π . We characterized the equality proving that the Whitneysphere is the only Lagrangian sphere6 with W(6) = 8π . This example can bedefined as the compactification of the generalized catenoid ofR4. In particular, theWhitney sphere is not only a Willmore surface but a minimum for the Willmorefunctional in its homotopy class. Using the classification of Ejiri, Montiel andMusso, and some results obtained by the authors in [4], we prove, in Corollary 1,that

The Whitney sphere is the only Willmore Lagrangian surface of genus zero inR4.

The rest of the paper deals with Willmore Lagrangian tori inR4. In some sense,the second part of this paper has been inspired by Pinkall’s paper. As we pointedout before, these Hopf tori are Willmore Lagrangian tori inR4 and, as they lie inthe unit sphere, are trivially invariant under an inversion centered at the origin. Inparticular, the property to be Lagrangian is kept under this inversion. This is ourstarting point to find new examples of Willmore tori. As a summary of Theorem 3:

We classify all the Willmore Lagrangian tori6 in R4 such that their imagesunder an inversion centred at a point outside6 (which are Willmore tori again)are also Lagrangian surfaces.

In the above classification, apart from the Willmore Hopf tori discovered by Pinkall,two new important families of examples appear.

The first one (see Examples B, Section 3) is constructed following Pinkall’sidea. We use theS1-fibration5:M3 ⊂ R4 −→ S1× R, where

M3 = {(z,w) ∈ C2− {0}/|z| = |w|}and5(z,w) = (zw/|zw|, log 2|zw|). The pull-back toM3 by 5 of a closed curveα in S1 × R, is a Lagrangian torusTα in C2 ≡ R4. The Willmore functional of

WILLMORE SURFACES OFR4 AND THE WHITNEY SPHERE 155

Tα is computed in terms of a functionalF on the curveα (see Proposition 1),andTα is a Willmore surface if and only ifα is a critical point ofF . The cor-responding Euler–Lagrange equation forF is computed in Proposition 2. Apartfrom the closed geodesics ofS1 × R, in Proposition 3 we obtain two nontrivialfamilies of closed curves inS1 × R which are critical points for the functionalF . The corresponding tori are characterized in Theorem 2 asthe only compactWillmore Lagrangian surfaces ofR4 coming from minimal surfaces in either thefour-dimensional sphere or the four-dimensional hyperbolic space. The first onemeans that these Willmore surfaces are minimal surfaces with respect to the spher-ical metric onR4. The second one means that the surface6 decomposes into threeparts6 = 6+ ∪ 60 ∪ 6−, lying, respectively, on the unit ball, on the unit sphere,and outside the closed unit ball, such that6± are minimal surfaces in hyperbolicspaces and60 is a set of totally geodesic points. So these examples are constructedin a similar way to the Willmore tori ofR3 obtained by Babich and Bobenko in [1].

The other family of Willmore surfaces appearing in Theorem 3, is a countablefamily of embedded Willmore tori (see Examples C, Section 3), which can be de-scribed in terms of a couple of circles (see Definition 1). For this family, we are ableto precisely determine the areas as well as the values of the Willmore functional(see Proposition 5), since we can control them very well by using elliptic functions.In fact, the o.d.e. that governs their local geometric behaviour is the sinh-Gordonequation.

2. Willmore Surfaces and the Whitney Sphere

Let R4 be four-dimensional Euclidean space and〈, 〉 the Euclidean metric on it.Given an immersionφ: 6 −→ R4 from a compact surface6, the Willmorefunctional ofφ is defined by

W(φ) =∫6

|H |2 dA,

whereH is the mean curvature vector ofφ and dA denotes the canonical measureof the induced metric, which will be also denoted by〈, 〉. The critical immersionsfor this functional are calledWillmore surfaces, and they are characterized by theequation (see [20])

1H − 2|H |2H + A(H) = 0,

where1 is the Laplacian on the normal bundle ofφ andA is the operator definedby

A(ξ) =2∑i=1

σ (ei, Aξ ei),

whereσ is the second fundamental form ofφ, Aξ the Weingarten endomorphismassociated to a normal vectorξ and {e1, e2} an orthonormal basis of the tangentbundle of6.


On the other hand, letC2 be two-dimensional complex Euclidean plane. Wedenote by〈, 〉 the Euclidean metric inC2 ≡ R4 and byJ a canonical complexstructure. The symplectic form� is defined by�(v,w) = 〈Jv,w〉 for vectorsvandw.

An immersionφ: 6 −→ C2 of a surface6 is calledLagrangianif any of thefollowing equivalent conditions are satisfied:

(i) φ∗� = 0,(ii) J (T6) = N6, whereT 6 andN6 are the tangent and normal bundles to6.

From this definition,J defines an isometry betweenT6 andN6 which commuteswith the tangent and normal connections. Also, the second fundamental formσ ofφ and the Weingarten endomorphismAξ associated to a normal vectorξ are relatedby σ (v,w) = JAJvw. So,〈σ (v,w), J z〉 defines a fully symmetric trilinear formon T6. If H denotes the mean curvature vector ofφ, the above properties joinedto the Codazzi equation imply thatJH is aclosed vector fieldon6.

Using the mentioned properties of a Lagrangian immersion, the above Willmoreequation can be written for Lagrangian immersions in the following way:

1H − (K + 2|H |2)H + 2σ (JH, JH) = 0, (1)

whereK is the Gauss curvature of6.Now we describe a result which will be use throughout the paper. In fact, all the

Willmore tori constructed in the paper belong to the following family of Lagrangiantori:

THEOREM A ([18]). Let φ: 6 −→ C2 be a Lagrangian immersion of a torus6. If X is a closed and conformal vector field on6 such thatσ (X,X) = ρJX

for certain functionρ, then the universal covering ofφ, φ: R2 −→ C2, is given byφ(t, s) = γ (t)β(s), whereγ :R −→ C∗ is a regular curve, andβ: R −→ S3 ⊂ C2

is a regular curve in the unit sphere, which is horizontal with respect to the Hopffibration5: S3 −→ S2. This means thatβ is a horizontal lift of a regular curve inS2.

If φ: 6 −→ C2 is a Lagrangian immersion of a genus zero compact surface6,it is well known thatφ is not an embedding, and so, using a result of Li and Yau[10], W(φ) ≥ 8π . In [4, corollary 5], the authors characterized the equality in theabove inequality, proving thatthe only compact Lagrangian surface of genus zerowith W(φ) = 8π is the Whitney sphere, which is defined by

(x, y, z) ∈ S2 7→ 1

1+ z2(x(1+ iz), y(1+ iz)) ∈ C2,

being

S2 = {(x, y, z) ∈ R3/x2+ y2+ z2 = 1}.


This example is not only a Willmore surface, but a minimum for the Willmorefunctional in its homotopy class. It is obtained by compactification of a complexcurve inC2 of finite total curvature−4π described by Hoffman and Osserman in[8], which can be written as

z ∈ C∗ 7→(z,

1

z

)∈ C2.

Now we are going to characterize this example, proving that the Whitney sphereis the only compact Willmore Lagrangian surface inC2 coming from a minimalsurface ofC2.

THEOREM 1. Letφ:6 −→ C2 be a Lagrangian immersion of a compact surface6. φ is the compactification by an inversion of a complete minimal surface of finitetotal curvature ofC2 if and only ifφ is the Whitney sphere.

Proof. Suppose thatφ is the compactification by an inversion of a completeminimal immersion with finite total curvatureφ: 6 −→ C2, where6 is6 punc-tured at a finite number of points. There is no restriction if we take the inversionF

asF : C2 ∪ {∞} −→ C2 ∪ {∞} defined byF(p) = p/|p|2. So, asφ = φ/|φ|2, avery easy computation of the mean curvature vector ofφ joint to the fact thatφ isa minimal immersion, says that

0= H + 2|φ|−2φ⊥, (2)

on 6, whereH is the mean curvature vector ofφ and⊥ means normal componentof φ. Deriving (2) and using the elementary properties of Lagrangian immersions,we obtain

0= 〈∇vJH,w〉 − 4|φ|−4〈dφ(v), φ〉〈dφ(w), Jφ〉 + 2|φ|−2〈σ (v,w), Jφ〉,where∇ is the Levi-Civita connection of the induced metric on6. As JH is aclosed vector field on6, the first term is symmetric. Also, since the third term issymmetric, we get that the second term is symmetric too, and so

δ ∧ δ = 0, (3)

whereδ andδ are the 1-forms on6 given by

δ(v) = 〈dφ(v), φ〉 and δ(v) = 〈dφ(v), Jφ〉.We include the following lemma because the next step in the proof will be used

in other parts of the paper.

LEMMA 1. Letφ: M −→ C2 − {0} be a Lagrangian immersion of a connectedsurfaceM. If δ and δ are the1-forms onM defined by

δ(v) = 〈dφ(v), φ〉 and δ(v) = 〈dφ(v), Jφ〉


and δ ∧ δ = 0, then there exist a closed and conformal vector fieldX onM andsmooth functionsa andb onM such thatφ is given byφ = aX + bJX, with a2+b2 = 1, where2a is the divergence ofX: divX. Moreover, the second fundamentalformσ of φ satisfies

σ (X,X) = ρJX, σ (X, V ) = −bJV,whereV is any orthogonal vector field toX andρ a smooth function onM.

Proof.Let

A = {p ∈ M; δp = 0} and B = {p ∈ M; δp = 0}.If A = M, thenφ = φ⊥. So the lemma follows takingX = Jφ⊥, a = 0 andb = 1.In this case,X is a parallel vector field. IfB = M, thenφ = φ> and the lemmafollows takingX = φ>, a = 1 andb = 0.

Let us assume now thatA andB are proper subsets ofM. Then, as|φ| has nozeroes, we have two disjoint closed subsetsA andB of M, such thatM/(A ∪ B)is a nonempty open subset ofM. Now, δ ∧ δ = 0 says that onM/A, we can writeδ = f δ for certain smooth functionf . TakingX = √1+ f 2 φ>, our immersionφis given onM/A by φ = aX+ bJX, wherea andb are smooth functions onM/Asatisfyinga2 + b2 = 1. Making a similar reasoning forB, we writeφ onM/Basφ = a′X′ + b′JX′ with a′2 + b′2 = 1. It is clear that, on the nonempty subsetM/(A∪B), we can takeX′ = X, a′ = a andb′ = b. In this way, we have obtaineda vector fieldX without zeroes and functionsa andb defined on the wholeM, suchthat

φ = aX + bJX, with a2+ b2 = 1.

The rest of the lemma can be found in [18, theorem 1], or can be easily checked.This finishes its proof. 2We continue with the proof of the theorem. Asφ⊥ = bJX and|φ|2 = |X|2, from(2) we get

H = −2|X|−2bJX. (4)

But, using the properties of the second fundamental form given in Lemma 1, themean curvature vector is also given by

2H = |X|−2(ρ − b)JX + cJv,for a certain functionc. Using (4), it follows thatρ + 3b = 0 andc = 0. Thenit is easy to check that the Gauss curvatureK of φ satisfies|H |2 = 2K. From [4,corollary 3], or [18, corollary 3], this means that our surface is either the Whitneysphere or totally geodesic. The compactness of our surface proves the theorem.2COROLLARY 1. The Whitney sphere is the only Willmore Lagrangian sphere ofC2.


Proof. We use the following result independently obtained in different waysby Ejiri, Montiel and Musso. Also, see [6] for a general reference about Penrosetwistor spaces.

THEOREM B ([7, 12, 14]).Letψ : 6 −→ S4 be a Willmore immersion of a sphere6. Then either the twistor lift ofψ (or the twistor lift of its antipodal−ψ) to thePenrose twistor spaceCP3 of S4 is a holomorphic curve orψ is the compactific-ation by the inverse of the stereographic projection of a complete minimal surfaceofR4 with finite total curvature and planar embedded ends.

In order to use this result, we are going to rewrite it when the ambient space isR4.LetG2

2 be the Grassmannian of oriented 2-planes inR4 andν: 6 −→ G22 the

Gauss map of an immersionφ: 6 −→ R4 from an oriented surface6. It is wellknown thatG2

2 = S21(1/√

2)×S22(1/√

2) and fori = 1,2, letνi : 6 −→ S2(1/√

2)be theith component ofν.

The two twistor spaces overR4 can be identified withR4 × S2i , i = 1,2, with

the natural projections, see [6]. The twistor liftsφi of φ to R4 × S2i are given by

φi = φ × νi, i = 1,2.As the Penrose twistorial constructions (as well as the Willmore surfaces) are

invariant under conformal transformations, taking the stereographic projection asa conformal transformation fromS4 punctured at a point ontoR4, we have thatTheorem B can be rewritten as

Let φ: 6 −→ R4 be a Willmore immersion of a sphere6. Then either one ofthe componentsνi , i = 1,2, of the Gauss mapν of φ is holomorphic orφ isthe compactification by an inversion of a complete surface ofR4 with finite totalcurvature and embedded planar ends.

If the first happens, then corollary 4 in [4] says that the surface is the Whitneysphere. If the latter happens, Theorem 1 finishes the proof. 2

3. Examples of Willmore Lagrangian Tori

In this section, we will construct new examples of Willmore tori inR4, which willbe characterized in Section 4. In order to make the paper self-contained, we startby describing the family of Hopf tori constructed by Pinkall in [16].

EXAMPLE A. Willmore Hopf tori: Let5: S3 −→ S2 be the Hopf fibration, whichis defined by

5(z,w) = (2zw, |z|2− |w|2),


whereS3 = {(z,w) ∈ C2/|z|2 + |w|2 = 1}. Let β be a closed curved inS2 and6 = 5−1(β). Then6 is a torus inS3 called aHopf torus. Pinkall [16] studied theseHopf tori, and among other things, he proved that ‘The Hopf torus6 = 5−1(β) isa Willmore surface ofC2 (or a Willmore surface ofS3) if and only ifβ is an elasticain S2’. This means thatβ is a critical point for the functional

∫ L0 (k

2+ 1)ds, wheres is the arc parameter ofβ, k its curvature andL its length. The Euler–Lagrangeequation of this functional, 2k′′ + k3+ k = 0, was obtained by Langer and Singer[9], who studied in depth this kind of curve, describing the closed elasticae ofS2

precisely. The corresponding tori will be referred to asWillmore Hopf tori. Finally,we remark that these Willmore Hopf tori are Lagrangian surfaces inC2.

EXAMPLE B. Let

M3 = {(z,w) ∈ C2− {0}/|z| = |w|}.It is easy to check thatM3 is a minimal hypersurface ofC2. We consider the actionof S1 onM3 defined by

eiθ (z, w) = (eiθ z,e−iθw).This action defines aS1-submersion5: M3 −→ S1 × R on the right circularcylinder given by

5(z,w) =(zw

|zw|, log 2|zw|).

We define a connection on this bundle by assigning to eachx ∈ M3 the subspaceof TxM3 orthogonal to the fibre of5 throughx. So we can talk about horizontallifts of curves ofS1× R. The horizontal lift of a curve ofS1 × R toM3 is uniqueup to isometries ofM3 of type{diag(eia,e−ia), a ∈ R}. Thus, given a curve

α = (α1, α2, α3): R −→ S1× R,its horizontal lift toM3 can be chosen by

t ∈ R 7→ 1√2(γ (t), γ (t)) ∈M3,

where γ is given by γ 2 = eα3(α1, α2). We remark thatα is parametrized by|α′(t)| = 2 if and only ifγ is parametrized by|γ ′(t)| = |γ (t)|.

Now, if α is a closed curve inS1 × R, 5−1(α) is a torus ofM3 ⊂ C2 and itsuniversal covering is defined by

φ(t, s) = γ (t)√2(eis ,e−is ),

whereγ 2 = eα3(α1, α2).We are going to study the properties of these tori in the following proposition.


PROPOSITION 1. Letα = (α1, α2, α3): [0, L/2] −→ S1 × R be a closed curveof lengthL parametrized by|α′(t)| = 2. Then:

(i) the torusTα = 5−1(α) is a Lagrangian surface ofC2;(ii) the image ofTα under the inversion ofC2, p 7→ p/|p|2, is Tα whereα =

(α1, α2,−α3);(iii) the torusTα is a Willmore surface ofC2 if and only ifα is a critical point for

the functional

F (α) =∫ L/2

0

(k2+ 1− α

′3

2

4

)dt,

wherek denotes the curvature of the cylindric curveα.

Remark 1.It is suitable to point out that the curvaturek and the componentα3

of a curveα in the cylinder parametrized by|α′| = 2 are related by

2k = (arcsin(α′3/2))′.

Proof.We recall that the universal covering of the torusTα is given by

φ(t, s) = γ (t)√2(eis ,e−is ),

whereγ 2 = eα3(α1, α2). We observe that asα is parametrized by|α′(t)| = 2, then|γ ′(t)| = |γ (t)| and then the induced metric byφ is |γ |2(dt2 + ds2). In particular,φ is a conformal immersion.

Now (i) and (ii) are easy to check.To prove (iii), we start computing the mean curvature vectorH of φ. It is

straigthforward to see that

2H =(kγ

|γ | +〈γ ′, J γ 〉|γ |4

)Jφt ,

wherekγ is the curvature ofγ .Hence, the Willmore functional is given by

W(Tα) =∫F

|γ |24

(kγ + 〈γ

′, J γ 〉|γ |3

)2

dt ds

= π

2

∫ L/2

0

(|γ |kγ + 〈γ

′, J γ 〉|γ |2

)2

dt,

beingF a fundamental region for the coveringR2 −→ Tα.Now, it is easy to check that the curvature ofα is

k = 12

(|γ |kγ − 〈γ

′, J γ 〉|γ |2

). (5)


So, the Willmore functional is written as

W(Tα) = 2π∫ L/2

0

(k + 〈γ

′, J γ 〉|γ |2

)2

dt.

Taking into account that the Laplacian ofα3 is

α′′3 = −4k〈γ ′, J γ 〉|γ |2 ,

and that our curveα is closed, the Willmore functional can be written as

W(Tα) = 2π∫ L/2

0

(k2+ 〈γ

′, J γ 〉2|γ |4

)dt.

Remembering now thatα is parametrized by|α′(t)| = 2, it is easy to see that

|γ |−4〈γ ′, J γ 〉2 = 1− α′3

2

4,

and soW(Tα) = 2πF (α).On the other hand, the mean curvature vectorsH andH ′ of Tα in C2 andM3

respectively, satisfyH = H ′ and then

W(Tα) =∫Tα

|H |2 dA =∫Tα

|H ′|2 dA.

Hence, the principle of symmetric criticality [15], can be applied here. In conclu-sion,Tα is critical for the Willmore functional if and only ifα is critical for thefunctionalF . This proves (iii). 2In the following result, we compute the Euler–Lagrange equation for the functionalF introduced in Proposition 1(iii).

PROPOSITION 2.Letα = (α1, α2, α3): [0, L/2] −→ S1×R be a closed curve oflenghtL parametrized by|α′(t)| = 2. Then,α is a critical point for the functional

F (α) =∫ L/2

0

(k2+ 1− α

′3

2

4

)dt

if and only if the curvaturek of α satisfies

k′′ + 2k3+ 2k

(1− 3α′3

2

4

)= 0. (6)


Proof. In [9], Langer and Singer deduced the Euler–Lagrange equation for thefunctional

F1(α) =∫ L/2

0(k2+ 1)dt

and they called a curve which is a critical point for this functionalelastica. Usingtheir result and the fact that the cylinder is a flat surface, we have that the firstderivative of the functionalF1(α) is given by

δF1(α) =∫ L/2

0

(k′′

2+ k3− k

)〈α′, δα〉dt,

whereδα stands for the variation vector field, andα = (−α2, α1, α3).Following their notation, it can be proved that the first derivative for the func-

tional

F2(α) = 14

∫ L/2

0α′3

2 dt

is given by

δF2(α) =∫ L/2

0k

(3α′3

2

4− 2

)〈α′, δα〉dt.

This is enough to finish the proof. 2The parallels of the right circular cylinder, that is,

αµ(t) = (cos 2t, sin 2t, µ), µ ∈ R,are trivially solutions of (6). The corresponding Willmore toriTαµ are, up to dila-tions inC2, the Clifford torus, which corresponds withµ = 0. In the next propos-ition, two nontrivial families of closed curves are constructed, which are criticalpoints for the functionalF .

PROPOSITION 3. (i)Letαλ: R −→ S1× R, λ ≤ 1, be the curve given by

αλ(t) =(e2iλ

∫ t0 cosh2 u(s) ds,2u(t)

),

whereu(t) is the solution to the o.d.e.u′2+ λ2 cosh4 u = 1 satisfyingu′(0) = 0.(ii) Letαν: R −→ S1× R, ν > 0, be the curve given by

αν(t) =(e2iν

∫ t0 sinh2 u(s) ds,2u(t)

),

whereu(t) is the solution to the o.d.e.u′2 + ν2 sinh4 u = 1 satisfyingu′(0) = 0.Then:


(a) there exist countably manyλ’s (respectively countably manyν’s) such thatαλ(respectivelyαν) is a closed curve;

(b) αλ andαν are critical points of the functionalF defined in Proposition 2, i.e.the curvatures ofαλ andαν verify Equation (6).

Proof. First, we note that the above parametrizations ofαλ and αν satisfy|α′λ(t)| = |α′ν(t)| = 2. Thus, it is an exercise to check that the curvatures ofαλandαν are given bykλ = λ sinhu coshu andkν = ν sinhu coshu and satisfy (6) inboth cases.

On the other hand, we claim thatu(t) is always a periodic function. This canbe proved, for example, by ensuring that the orbitst 7→ (u(t), u′(t)) are closedcurves. By another reasoning, if we writey = tanhu, theny satisfies the differ-ential equationy′2 = (1− y2)2 − λ2 in case (i) (resp.y′2 = (1− y2)2 − ν2y4 incase (ii)) that can be solved by standard techniques in terms of elliptic functions[5]. This leads us to conclude that there existT = T (λ) andT = T (ν) such thatu(t + 2T ) = u(t) = u(−t), ∀t ∈ R.

To determine whichαλ’s andαν ’s are closed curves, we start callingB(t) =2λ∫ t

0 cosh2 u(s)ds in case (i) (resp.B(t) = 2ν∫ t

0 sinh2 u(s)ds in case (ii)). Then,for anym ∈ Z, B(t + 2mT ) = B(t) + 2mB(T ) from the above properties ofu(t). Hence, we must consider the rationality conditionB(T )/π ∈ Q. So we needquantitative information aboutB(T ) that we are going to obtain from the explicitexpression ofB(T ) in terms of the Heuman-lambda function [3].

We denote by sn, cn and dn the elementary Jacobian elliptic functions [3]. Incase (i), we arrive at

cosh2 u(t) = dn2(r t, k)

λ(1+ k2sn2(r t, k)),

wherer = √λ+ 1 andk2 = (1− λ)/(1+ λ). Then, using formula 410.04 of [3],we compute thatB(T )/π = 30(π/4, k), which increases monotonically from 1/2to√

2/2 asλ increases from 0 to 1.In case (ii), we must distinguish another two cases: ifν ≤ 1 we arrive at

sinh2 u(t) = cn2(r t, k)

ν(1+ sn2(r t, k)),

wherer = √ν + 1 andk2 = (1 − ν)/(1 + ν). Then, using formula 411.03 of[3], we compute thatB(T )/π = (1−30(π/4, k)), which decreases monotonicallyfrom 1/2 to 1−√2/2 asν increases from 0 to 1.

If ν ≥ 1 we arrive at

sinh2 u(t) = cn2(r t, k)

ν + sn2(r t, k),

wherer = √2ν andk2 = (ν − 1)/(2ν). Then, again using formula 411.03 of[3], we compute thatB(T )/π = 1− 30(sin−1√ν/(ν + 1), k), which decreasesmonotonically from 1−√2/2 to 0 asν increases from 1 to∞. 2


We are now able to characterize the corresponding Willmore tori associated to thecurves given in Proposition 3. In Theorem 1, we studied the Willmore Lagrangiancompact surfaces ofC2 coming from the minimal surfaces ofC2. Similarly, in thenext theorem, we will describe the Willmore Lagrangian compact surfaces ofC2

coming from minimal surfaces of the sphere or the hyperbolic space.

THEOREM 2. Let φ: 6 −→ C2 be a Lagrangian immersion of an orientablecompact surface6.

(i) If φ:6 −→ C2 is a minimal immersion with respect to the spherical metric onC2, g = 4(1+ |p|2)−2〈, 〉, then, up to dilations,φ is congruent to some Will-more torusTα, whereα is one of the closed curves given in Proposition3(i).

(ii) If 6 = {p ∈ 6/|φ(p)| 6= 1} andφ: 6 −→ C2− S3 is a minimal immersionwith respect to the hyperbolic metric onC2− S3, g = 4(1− |p|2)−2〈, 〉, then,up to dilations,φ is congruent to some Willmore torusTα, whereα is one ofthe closed curves given in Proposition3(ii).

Remark 2.(C2,4(1+ |p|2)−2〈, 〉) is isometric toS4 punctured at a point withits canonical metric of constant curvature 1.

The bounded connected component ofC2−S3 with the metric 4(1−|p|2)−2〈, 〉is the four-dimensional hyperbolic space of constant curvature−1, and the otherconnected component is isometric, by an inversion, to the four-dimensional hy-perbolic space of constant curvature−1 punctured at some point. So this secondfamily of tori is constructed using a similar method to that used by Babich andBobenko in [1].

Proof of (i). As a minimal surface ofS4 is a Willmore surface and this propertyis kept under conformal transformations, our immersionφ is a Willmore immersioninC2. Also, computing the mean curvature vector ofφ with respect to the sphericalmetric and using the minimality ofφ, we get

0= H + 2

1+ |φ|2 φ⊥, (7)

whereH is the mean curvature vector ofφ and⊥ means normal component toφ.Now we will proceed in a very similar way as in the proof of Theorem 1.

Deriving (7), we obtain

0= 〈∇vJH,w〉 − 4

(1+ |φ|2)2〈dφ(v), φ〉〈dφ(w), Jφ〉 +2

1+ |φ|2 〈σ (v,w), Jφ〉,

where∇ is the Levi-Civita connection of the metricφ∗〈, 〉. Making the same reas-oning as in the proof of Theorem 1, the second term of the above expression issymmetric, and soδ ∧ δ = 0, whereδ andδ are the 1-forms on6 given by

δ(v) = 〈dφ(v), φ〉 and δ(v) = 〈dφ(v), Jφ〉.


From Lemma 1, we conclude that there exist a closed and conformal vector fieldX on6, without zeroes, and functionsa andb defined on the whole6 such that

φ = aX + bJX, a2 + b2 = 1.

In particular, as6 admits a conformal vector field without zeroes,6 is a torus.Sinceφ⊥ = bJX and|φ| = |X|, from (7) we get

H = −2(1+ |X|2)−2bJX. (8)

But, using the properties of the second fundamental form ofφ given in Lemma 1,we also obtain that

2H = |X|−2(ρ − b)JX + cJV,for a certain functionc, whereV is a unit vector field orthogonal toX. Comparingthis expression ofH with that given in (8), we have

(1+ |X|2)ρ = (1− 3|X|2)b, c = 0. (9)

Deriving φ = aX + bJX, using Lemma 1 and (9), it is easy to check that thegradient ofb satisfies

∇b = a b(|X|2− 1)

|X|2(|X|2+ 1)X.

But ∇|X|2 = 2aX becauseX is a closed and conformal vector field. Thus thegradient of the functionb|X|2(1+ |X|2)−2 is zero and that function is a constantµ. Hence, from (9) we get

b = µ (1+ |X|2)2

|X|2 , ρ = µ (1+ |X|2)(1− 3|X|2)|X|2 .

As 2a = divX andb = 4µ cosh2(log |X|), the equationa2 + b2 = 1 becomes, inthe following o.d.e.,(

divX

2

)2

+ 16µ2 cosh4(log |X|) = 1. (10)

On the other hand, from Theorem A, the universal covering ofφ, φ: R2 −→C2 is given byφ(t, s) = γ (t)β(s), whereγ : R −→ C∗ is a regular curve andβ: R −→ S3 is the horizontal lift by the Hopf fibration of a regular curveβ inS2. As c = 0, the curvature ofβ is zero and soβ is a great circle inS2. Then ourimmersion can be written as

φ(t, s) = γ (t)√2(eis ,e−is ).


Hence, the immersionφ is exactly5−1(α) whereα is the cylindric curveα =(γ 2/|γ |2, log |γ |2). If X is the corresponding closed and conformal vector field onR2 andu the function onR2 defined by eu = |X| = |γ |, then Equation (10) isrewritten as

u′2+ 16µ2 cosh4 u = 1. (11)

Moreover, asX = φt , the equationφ = aX+ bJ X on the points(t,0) becomes inγ ′ = (a − ib)γ , and soγ is given by

γ (t) = e∫ t

0 (a−ib)(r) dr = eu(t)

eu(0)e4µi

∫ t0 cosh2 u(r) dr.

As a consequence, the curveα is written as

α(t) = (e8µi∫ t

0 cosh2 u(r) dr,2u(t)),

whereu(t) is a solution of Equation (11). So our curve corresponds to that givenin Proposition 3(i) withλ = 4µ.

Proof of (ii). By a similar reason to (i),6 satisfies Equation (1). Using thatEquation (1) is elliptic and following a similar reasoning to that used by Babichand Bobenko in the proof of theorem 4 of [1], it follows that on60 = φ−1(S3)

Equation (1) is also satisfied. So, our surface6 is a Willmore surface ofC2. Fol-lowing again the proof of theorem 4 of [1], it can be proved in a similar way that60 is a set of umbilical points and then, by analyticity,60 has an empty interior.

Also, computing the mean curvature vector ofφ with respect to the hyperbolicmetric and using the minimality ofφ, we get on6

H = 2

(1− |φ|2)φ⊥,

whereH is the mean curvature vector ofφ and⊥ means normal component.Using the same reasoning as in (i), we get that the 1-formsδ and δ (which are

defined on the whole6) satisfy on6, δ∧ δ = 0. As60 has an empty interior, thenby continuityδ ∧ δ = 0 on the whole6.

Repeating the reasoning made in the proof of (i), we getφ = aX+bJX, whereX is a closed and conformal vector field on6 anda2 + b2 = 1. So6 must be atorus. Also, the functionsb andρ are given in this case by

b = µ(1− |X|2)2

|X|2 , ρ = µ(1− |X|2)(1+ 3|X|2)|X|2 ,

on 6, and by continuity on the whole6. In particular,60 is a set of geodesicpoints.

Using the same arguments of (i), ifφ: R2 −→ C2 is the universal covering ofφ, thenφ is 5−1(α) whereα is the cylindric curve

α(t) = (e8µi∫ t

0 sinh2 u(r) dr,2u(t)),


u being a solution of the equation

u′2+ 16µ2 sinh4 u = 1. (12)

So our curve corresponds to that given in Proposition 3(ii) withν = 4λ. 2EXAMPLE C. If we consider the curveβ: R −→ S3 given by

β(s) = (cosψ e−i tanψ s, sinψ ei cotψ s),

with ψ ∈ (0, π/2), then it is clear thatβ = 5 ◦ β is the parallel ofS2 of latitudeπ/2−2ψ , where5: S3 −→ S2 is the Hopf fibration. Also we have that〈β ′, J β〉 =0. This just means thatβ is a horizontal lift, by the Hopf fibration, of the parallelβ in S2. It is unique up to rotations inS3. Heres is the arc parameter forβ and wenote that|β ′(s)| = 2.

We observe thatβ is a closed curve inS3 if and only if tan2ψ ∈ Q.

DEFINITION 1. For anym,n ∈ N, 0< n < m, (n,m) = 1, let

φm,n: R2 −→ C2

be the immersion defined by

φm,n(t, s) = γ (t)β(s),whereγ is the regular curve inC∗ parametrized by|γ (t)| = |γ ′(t)| such thatγ 2 isthe circle centred at((m+ n)/(m− n),0) with radius 2

√mn/(m− n) andβ is the

horizontal lift of the parallel inS2 of latitude arcsin(m− n)/(m+ n). According tothe above expression,

β(s) = 1√m+ n

(√me−i

√n/m s,

√n ei√m/n s

).

Thanks to the properties of periodicity that we are going to prove in the nextProposition, we can introduce a new family of Willmore Lagrangian tori.

PROPOSITION 4. The immersionsφm,n: R2 −→ C2 described in Definition1are doubly periodic with respect to the rectangular lattice3m,n in the(t, s)-planegenerated by the vectors(4

√mnK/(

√m+√n)2,0) and(0,2

√mnπ), whereK is

the complete elliptic integral of first kind with modulus8√mn(m+n)/(√m+√n)4

[3].The toriTm,n = R2/3m,n are Willmore Lagrangian embedded surfaces inC2.Proof. First, it is easy to check that our immersionsφm,n are conformal, since

the induced metric byφm,n is written as e2u(t)(dt2 + ds2), where e2u(t) = |γ (t)|2.For our purposes, we need the explicit expression ofγ . This necessarily involvesthe use of Jacobian elliptic functions [3].


Claim. The curveγ determined in Definition 1 is given by

γ (t) = exp

[u(t)+ (m− n)i

2√mn

∫ t

0sinh 2u(x)dx

],

whereu = u(t) is the solution to the sinh-Gordon equation

u′′ + (m− n)2

4mnsinh 4u = 0

with initial conditionsu′(0) = 0, sinh 2u(0) = 2√mn/(m− n).

In fact, in order to simplify the notation, let

a = (√m+√n)2/(m− n) and b = (√m+√n)2/2√mn.We easily obtain that|γ | = |γ ′| and that the curvature ofα = γ 2 is constant,

exactly(m − n)/2√mn. Moreover, we also need the explicit expression foru =u(t)which is given by e2u(t) = a dn(b t), where dn is the ‘delta amplitude’ Jacobianelliptic function [3] with modulus 1− 1/a4 (a = e2u(0)). Now, it is not difficult todeduce that the farthest and nearest points from the origin to the circleα areα(0) =(a,0) andα(K/b) = (1/a,0). The middle point of them is just(cosh 2u(0),0) =((m+ n)/(m− n),0), the center ofα. So the claim is true.

If we write

F(t) =∫ t

0e2u(x) dx and G(t) =

∫ t

0e−2u(x) dx,

we can get thatF(t) = arcsin(sn(b t)), if t ∈ (−K/b,K/b) andG(t) = arccos(cd(b t)), if t ∈ (0,2K/b), where sn and cd are Jacobian elliptic functions [3].From here, it is easy to prove that∫ t+2K/b

0sinh 2u(x)dx =

∫ t

0sinh 2u(x)dx

and thenγ (t + 2K/b) = γ (t), ∀t ∈ R.Looking at Definition 1, it is clear thatβ(s + 2

√mnπ) = β(s), ∀s ∈ R.

On the other hand,φm,n is a Lagrangian immersion because the curveβ ishorizontal.

To prove thatφ: Tm,n −→ C2 is an embedding, it is enough to notice thatγ (t),t ∈ [0,2K/b], andβ(s), s ∈ [0,2√mnπ ], are simple closed curves inC∗ andS3,respectively.

Finally, since we knowφm,n explicitly, it is a very long but straightforwardcomputation to check that Equation (1) is satisfied for the toriTm,n. 2PROPOSITION 5. The Willmore Lagrangian toriTm,n = R2/3m,n introduced inProposition4 satisfy the following properties:


(i) the areaA of the torusTm,n is

A = 4mn

m− nπ2;

(ii) the Willmore functionalW of the torusTm,n is

W = 2π

((√m+√n)2E + (m− n)2

(√m+√n)2K

),

whereK andE (depending onm andn) are the complete elliptic integrals offirst and second kind with modulus8

√mn(m+ n)/(√m+√n)4, respectively

[3].(iii) the torusTm,n is invariant under the inversion ofR4 given byp 7→ p/|p|2.

Proof.We follow the same notation as in the proof of Proposition 4. Using thatthe measure of the induced metric of the torus is given by dA = e2u(t) dt ds andthatF(2K/b) = 2

√mnπ/(m− n), we obtain the formula of the areaA.

For Willmore’s we first need to check that|H |2 = k2αe

2u + k2β e−2u. Taking into

account that∫ 2K/b

0e4u(t) dt = (a2/b)

∫ 2K

0dn2w dw = (2a2/b)E

and thatkβ = (m− n)/(2√mn), we finally get the expression given forW .We first observe thatφ/|φ|2 = (γ /|γ |2)β. But γ 2 is a circle with centerC =

((m + n)/(m− n),0) and radiusR = 2√mn/(m − n). Then|C|2 − R2 = 1 and

this just implies that it remains invariant under the inversion. 2

4. Classification

In this section, we are going to characterize the families of Willmore tori defined inthe last section among the compact Willmore surfaces ofC2. We will assume thatthe Willmore surface is Lagrangian, and also that its image under some inversion,which is a Willmore surface again is, in addition, a Lagrangian surface.

THEOREM 3. Letφ: 6 −→ C2 be a Willmore immersion of an orientable com-pact surface6 such thatφ and its image under some inversion ofC2, centered at apoint ofC2−{φ(6)}, are both Lagrangian immersions. Thenφ is congruent, up todilations ofC2, either to a Willmore Hopf torus, or to someTα whereα is a closedcurve in the right circular cylinder, critical for the functionalF (see Propositions1and2), or to someTm,n,m,n ∈ N, n < m, (m, n) = 1 (see Proposition4).


Proof.Up to a translation, we can assume that the inversion of the hypothesis ofthe theorem isF : C2 ∪ {∞} −→ C2 ∪ {∞} defined byF(p) = p/|p|2. Let

ψ = F ◦ φ = φ

|φ|2 : 6 −→ C2,

which is also a Lagrangian immersion by assumption. Ifδ andδ are the 1-forms on6 given by

δ(v) = 〈dφ(v), φ〉 and δ(v) = 〈dφ(v), Jφ〉,then it is easy to see thatψ∗� = (−2/|φ|6)(δ ∧ δ); asψ is also Lagrangian, weobtain that

δ ∧ δ = 0. (13)

From Lemma 1 and Theorem A, we get that the universal covering ofφ, which willbe also noted byφ, φ: R2 −→ C2, is given by

φ(t, s) = γ (t)β(s),whereγ : R −→ C∗ is a regular curve andβ: R −→ S3 ⊂ C2 is a regular curve inthe unit sphere, which is a horizontal lift (with respect to the Hopf fibration) of aregular curveβ in the sphereS2.

From now on, we reparametrizedγ by |γ ′(t)| = |γ (t)| and β by |β ′(s)| =1. Then the induced metric is given by|γ |2(dt2 × ds2) and, in particular,φ is aconformal immersion.

Our first purpose is to write down in a suitable way the equation of a Willmoresurface (Equation (1)) in terms ofγ andβ.

It is straightforward to see that the second fundamental formσ of φ is given by

σ (∂t , ∂t ) = |γ |kγ Jφt ,

σ (∂s, ∂s) = 〈γ′, J γ 〉|γ |2 Jφt + 2kβJφs,

wherekγ andkβ are the curvatures of the planar curveγ and the spherical curveβ,respectively. So, the mean curvature vectorH of φ is given by

2H =(kγ

|γ | +〈γ ′, J γ 〉|γ |4

)Jφt + 2kβ

|γ |2Jφs.

It will be useful for our purpose to introduce the curveα in C∗ defined byα(t) = γ 2(t). We observe that the parametrization ofα satisfies|α′| = 2|α|. Theircurvatures are related by

2kα = kγ

|γ | +〈γ ′, J γ 〉|γ |4 .


Thus, the mean curvature vector is

H = kαJφt + kβ

|α|Jφs.

Now, a very long but really straightforward computation says thatφ is a Will-more surface, i.e.φ satisfies Equation (1), if and only if the following equationshold:

|α| k′′α + |α|′ k′α +〈α′, J α〉2|α|3 kα − 3〈α′, J α〉 k2

α + 2|α|3 k3α−

− (log |α|)′ kβ + 2

( 〈α′, J α〉2|α|2 − |α|kα

)k2β = 0, (14)

kβ + 2k3β − 2kβ

(|α|2k2

α −〈α′, J α〉|α| kα

)= 0,

where′ (respectively.) means derivative with respect tot (respectivelys).In order to manipulate the above equations, we consider on the cylinderS1×R

the curveα = (α1, α2, α3) defined byα = (α/|α|, log |α|). We observe that|α′| =2. Its curvaturekα can be easily computed, obtaining that

kα = |α|kα − 〈α′, J α〉

2|α|2 . (15)

Using (15), Equations (14) can be changed into the following easier equations:

k′′α + 2k3α + 2kα

(1− 3α′3

2

4− k2

β

)− α′3 kβ = 0,

kβ + 2k3β − 2kβ

(k2α − 1+ α

′32

4

)= 0. (16)

Now, deriving the second equation of (16) with respect tot , we obtain

kβ

(k2α − 1+ α

′32

4

)′= 0.

So eitherkβ ≡ 0 or k2α − 1+ (α′32

/4) is a constantρ.

Case 1.Let us assumekβ ≡ 0.Thenβ is a great circle inS2 and, up to rotations, its horizontal lift toS3 is givenby β(s) = (1/√2)(eis ,e−is ). So our original immersionφ is given by

φ(t, s) = γ (t)√2(eis ,e−is ).


With the notation of Example B, it is clear thatφ(R2) lies inM3 and using thefibration 5, it is also clear thatφ and the universal covering of5−1(α) are thesame. Equations (16) become

k′′α + 2k3α + 2kα

(1− 3α′3

2

4

)= 0.

This equation is just Equation (6) in Proposition 2. Hence, our torus, up to dilationsof C2, is one of the given in Example B.

Case 2.Suppose now that

k2α − 1+ α

′3

2

4≡ ρ. (17)

Deriving (17) with respect tot , we obtain

kαk′α +

α′3α′′3

4= 0. (18)

But deriving (15), we get

k′α − α′3〈α′, J α〉

2|α|2 = |α|k′α . (19)

Using

α′′3 = −2kα |α|−2〈α′, J α〉, (20)

Equations (18) and (19) become|α|kαk′α = 0. From here, asα is a curve inC∗, itis clear that eitherkα = 0 orkα is constant.

Case 2.1.If kα = 0,α must be a closed geodesic inS1×R because our surface is atorus. Necessarily this implies thatγ must be a circle centered at the origin ofC2.Hence, up to dilations ofC2, we can assume thatγ is a circle of radius one. In thiscase, our torus is5−1(β), where5: S3 −→ S2 is the Hopf fibration, and using thesecond equation of (16),β is a curve inS2 satisfyingkβ + 2k3

β + 2kβ = 0. In thiscase, our torus is one of the Willmore Hopf tori described by Pinkall in [16]. Weobserve that this equation corresponds exactly with the one given in Example Ataking into account that the arc parameter forβ is 2s.

Case 2.2.The last case to study is whenkα is constant. In this case, it follows from(19) that

k′α = α′3〈α′, J α〉

2|α|2 .


Deriving this equation and using (15) and (20), we getk′′α = 2(α′32 − 2)kα.

Putting this equation, joined to (17) in Equations (16), we obtain

α′3kβ + 2kα(k2β − ρ) = 0, kβ + 2kβ(k

2β − ρ) = 0. (21)

Deriving the first equation of (21) with respect tos and again using both equationsof (21), we conclude that(

k2α +

α′32

4

)kβ (k

2β − ρ) = 0.

But from (17)k2α + α′32

/4 = 1+ ρ. If 1 + ρ = 0, thenkα = 0 andα3 is constant.Hence, our surface is, up to dilations, the Clifford torus which is the simplestWillmore Hopf torus. So we can assume that 1+ ρ > 0. As in this casekβ isnonnull, the above equation says thatk2

β = ρ. So the curveβ is a small circle ofS2.

On the other hand, sincekα is constant and our surface compact,α must be acircle inC∗ , say of radiusR and centerC. So|α − C| = R andkα = 1/R. Fromhere we arrive at

|α|2+ |C|2− 2〈α, C〉 = R2, α′ = 2|α|RJ(α − C). (22)

Taking into account thatk2β = ρ and kα = 1/R, using (15), (17) and the two

equations of (22), we can conclude that

R2(1+ k2β) = |C|2. (23)

Finally, as our surface is compact, the horizontal liftβ to S3 of our small circleβ of S2 must be a closed curve. Looking at Example C in Section 3, we havethat there exist positive integer numbersm,n with n < m and (m, n) = 1 suchthat kβ = (m − n)/2√mn. It is not a restriction to take the centerC of α likeC = (a,0). We obtain from (23) thata2 = R2(m + n)2/(4mn). This relation isinvariant under homotheties ofC and so, up to dilations onC2, we can takeC anda like in Example C. So in this last case, our torus is congruent with some torusTm,n described in Example C of Section 3. 2

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