minimal submanifolds of hyperbolic spaces via harmonic morphisms

Geometriae Dedicata 62: 269-279, 1996. 269 © 1996 KluwerAcademic Publishers. Printed in the Netherlands.

Minimal Submanifolds of Hyperbolic Spaces via Harmonic Morphisms

SIGMUNDUR GUDMUNDSSON Department of Mathematics, University of Lund, Box 118, S-221 O0 Lund, Sweden. e.mail: [email protected]

(Received: 23 February 1995)

Abstract. In this paper we give a method for constructing complete minimal submanifolds of the hyperbolic spaces H '~. They are regular fibres of harmonic morphisms from H m with values in Riemann surfaces.

Mathematics Subject Classification (1991): 58E20, 53C42.

1. Introduction

One of the classical problems in differential geometry is the study of minimal surfaces in a given Riemannian ambient space (M, 9). One case that has been studied very recently is that of minimal surfaces in 3-dimensional hyperbolic space, see for example [20], [21], [22] and [1].

The study of minimal surfaces is closely related to the theory of harmonic maps. This is due to the fact that they can locally be described by (weakly) conformal harmonic immersions ~b: (N 2, h) -+ (M, g) from a Riemann surface. For the dual situation of submersions we have the following result due to Baird and Eells, see [3]: If q~: (M, g) -+ (N 2, h) is a horizontally conformal submersion (see next section) to a surface, then the following conditions are equivalent:

• ~b is a harmonic map, • each fibre q5 -1 (z0) of ~b is a minimal submanifold of (M, 9).

This implies that harmonic morphisms or equivalently horizontally conformal harmonic maps are useful tools for constructing minimal submanifolds of codimension 2.

For harmonic maps there exist many interesting existence results which can, for example, be found in the excellent review articles [7], [8] and [17]. When studying harmonic morphisms the additional condition of horizontal conformality means that we are dealing with an overdetermined system of partial differential equations. For this situation there is no general theory of existence, not even locally. This makes the construction of concrete examples between specific manifolds of major importance.

270 SIGMUNDUR GUDMUNDSSON

Harmonic morphisms from the hyperbolic 3-space H 3 to a Riemann surface N 2 have been classified in [5] in terms of meromorphic functions on the surface. In [2] locally defined harmonic morphisms from H a to Riemann surfaces are classified in terms of Hermitian structures on the domain.

For hyperbolic spaces H "~ with m > 4 the only known examples of harmonic morphisms to Riemann surfaces are, up to a composition with a conformal map of the codomain, restrictions of the projections H ~ --+ H 2, H "~ --+ C and H m - {0} --+ C t_J { cc }. These maps are rather trivial in the sense that they are horizontally homothetic, have totally geodesic fibres and integrable horizontal distributions. For this see [11] where the author classifies such maps from hyperbolic spaces H m and other simply connected space forms.

In this paper we construct for m > 2 explicit globally defined harmonic morphisms from any odd-dimensional hyperbolic space H 2m+l to Riemann surfaces. Their regular fibres are complete non-totally geodesic minimal submanifolds of H 2m+1 of codimension 2. In each case they form a conformal foliation of an open and dense subset of H 2m+l.

2. Harmonic Morphisms

Throughout this paper we assume that all our objects such as manifolds, maps, etc., are smooth, i.e. in the C ~-category. Let M and N be manifolds of dimensions m and n, respectively.

DEFINITION 2.1. A map ¢: (M, g)--.(N, h) between Riemannian manifolds is called a harmonic morphism if for any harmonic function f:U --+ ]~ defined on an open subset U of N with ¢ - I ( u ) non-empty, f o ¢: ¢-1(U) ~ I~ is a harmonic function.

I f m < n then every harmonic morphism ¢: (M, g) --+ (N, h) is constant, hence not of interest, ff m >__ n then a non-constant harmonic morphism ¢: (M, g) --+ (N, h) is submersive outside the critical set C¢ := {x E M[ dCx = 0}, which has an open and dense complement M* := M - C¢. In [9] and [16] Fuglede and Ishihara independently characterize harmonic morphisms between Riemannian manifolds as those harmonic maps which are horizontally conformal in the following sense: At each point p E M* let ];p be the vertical space at p given by Vp := Ker dCp C 7pM and 7-/p : = V~ be the horizontal space. Here _1_ denotes the orthogonal complement with respect to the metric 9 on M. The map ¢: (M, g) --+ (N, h) is said to be horizontally (weakly) conformal if there exists a function A: M* ~ ~+ such that

A2g(X,Y) = h(d¢(X) , de(Y)) for all X , Y e 7-[.

In what follows we are mainly interested in maps ¢: (M, g) ~ (N 2, h) with values in a surface. In that case the condition of ¢ being a harmonic morphism only depends on the conformal structure of N 2. Hence if the surface is oriented we can

MINIMAL SUBMANIFOLDS OF HYPERBOLIC SPACES 271

take it to be a Riemann surface. Examples of harmonic morphisms are holomorphic maps between Riemann surfaces.

If E m is the standard m-dimensional Euclidean space and (N 2, h) is a Riemann surface, then the harmonicity and horizontal conformality of a map ¢: E "~ ~ N 2 are expressed as follows: For every complex coordinate z: U --+ C on the Riemann surface N 2 we have

o ¢)

-~ 02 : E

k=l

= 0 .

k=l

3. The Semi-Riemannian Case

In this paper we construct harmonic morphisms from the Riemannian m- dimensional hyperbolic space H m. Our method is to lift the problem into the (m + 1)-dimensional Minkowski space and solve it there. For this reason we now describe how the fundamental characterization of a harmonic morphism has been generalized to the semi-Riemannian case.

It is well known that the notions of a harmonic function, a harmonic map and that of horizontal (weak) conformality carry over to the case of semi-Riemannian manifolds. In [19] Parmar studies harmonic morphisms in this situation, i.e. maps ¢: (M, g) -+ (N, h) between semi-Riemannian manifolds which pull back harmonic functions on (N, h) to harmonic functions on (M, g). He shows that every harmonic map which is horizontally (weakly) conformal is a harmonic morphism, but fails to give a definite answer to whether the converse is true. Recently this open problem was completely solved by Fuglede in [10]. He proves the following:

THEOREM 3.1. A map ¢: (M, g) --+ (N, h) between semi-Riemannian manifolds is a harmonic morphism if and only if it is a horizontally (weakly) conformal harmonic map.

4. Hyperbolic Spaces

Let M "~+1 be the (m + 1)-dimensional Minkowski space, i.e. ~ + 1 equipped with the semi-Euclidean metric

m

(x, y) := -xoYo + ~ xkyk, k=l


where x = (xo, x l , . . . ,Xm).PutZ m+l : = {x E M~+ll(x,x) < 0 and x0 > 0} and let the Lie group ]R + act on Z m+l by multiplication, i.e. r: x ~ r • x. The quotient space of Z m+l under this action can be identified with the upper sheet of the m-dimensional hyperboloid

H ~ := {x E M~+Xl(x,x) = - 1 andxo > 0}.

It is well known that the restriction of the metric ( , ) to 11 m is positive definite and defines the hyperbolic space ( H m, ( , ) ) of constant sectional curvature - 1 . The natural projection 7r: Z '~+1 --+ H m onto the quotient space is given by

L E M M A 4.1. The natural projection 7r: Z m+l --+ 11 m is a submersive harmonic

morphism. Proof. If p E H m then the tangent space TpH "~ is given by TpH m = {v E

M ' ~ + l l ( v , p ) = 0}. For x E Z m+l the vertical space 12~ of 7r at x is simply Vx = Ker dTrx = {A • x E M'~+II~ E I~} and the horizontal space 7-/~ = 13~ can be identified with T~(x)H m. It is not difficult to see that if v E 7/~ then

OTto(v) = s o

1 (d~(v), a t ( w ) ) - ( x , x ) ( v ' w )

for all v, w E 7~x. This means that the map 7r is a horizontally conformal submersion.

If i: H m ~ Z ra+l is the canonical isometric embedding of H m into Z m+l then

the tension field of i o 7r satisfies

02(i° )

- - Ox2o + o2(io ) m ( ion) .

This implies that for each x E Z re+l, r ( i o r ) ( x ) lies in the orthogonal complement (Tio~(x)H~) ± . It then follows from Lemma 1.3.6 of [18] that 7r is a harmonic map. []

THEOREM 4.2. Let (9: U ~ N 2 be a map defined on an open subset U of_ti m with values in a Riemann surface. Let 7r: Z ra+l ~ H m be the natural projection,

be the pre-image 7r-l( u ) o f Tr and ~: (J --+ N 2 be the composition ¢ := (~ o 7r [~. Then the following conditions are equivalent:

• ~ is a harmonic morphism, • ~ is a harmonic morphism.


Proof This is a semi-Riemannian version of Proposition 4.3 in [13]. Since the map 7r is submersive the same proof can be used here. []

A map ~ : ~ --+ N z from an open subset U of Z m+I is said to be R+-invadant when: x E U implies that Az E U and q~(Az) = ~(z) for all A E ~+. Theorem 4.2 shows that there is a bijective correspondence between harmonic morphisms from the m-dimensional hyperbolic space ( / / r a ( , ) ) and/1~ +-invariant maps from open subsets U" of Z ~+I satisfying the following conditions:

.,-(; o ; ) o2(; o o2(z o - - + _ 0 ,

k = l

(1)

- k= ,L J (2)

for every complex coordinate z: V --+ C on N 2. Let (N 2, h) be a Riemann surface with a compatible Riemannian metric h.

Equip the product manifold N 2 x Z ~+1 with the product metric 9 := h × (,). For a map q/: N 2 × Z ~+1 -+ E2 with values in a Riemann surface E2 and points w E N 2 , p E Z ra+l w e denote by ~ : Z m+l --+ ~2 and ~ p : N 2 -+ ~2

the maps given by ~w:q ~ ~ ( w , q ) and ~Pv:z ~ ~ ( z , p ) . The map ~: ( N2 × Z'~+I, 9) -+ E2 is said to be a harmonic morphism in each variable separately if ~ : (Z ~+1 , ( , ) ) --+ ~2 and ~p: (N 2, h) --+ E2 are harmonic morphisms for all w E N 2 and p E Z m+l •

PROPOSITION 4.3. Let ~: N 2 × Z m+l --+ ~2 be a harmonic morphism in each variable separately. I f zo E ~2 and d~l ¢ 0 on ~- l ( zo ) , then any smooth local solution ~: U --+ N 2, w = (9(p) to the equation

• ( w , p ) : zo,

defined on an open subset U of Z re+l, is a harmonic morphism. Proof The proof of this result works exactly the same way as for the Rieman-

nian version given in Proposition 2.5 of [15]. []

5. Totally Geodesic Fibres

In this section we construct explicit harmonic morphisms from hyperbolic spaces / / '~ with values in a Riemann surface. The reader should note that the regular fibres of the constructed maps all lie on some linear (m - 1)-dimensional subspace of


the Minkowski space M m+l so they are totally geodesic in H "~. For geometrically more interesting minimal submanifolds of H 2m+l see the last section.

Let IV 2 be a Riemann surface and let f = (f0, f l , . . . , f ~ ) : N 2 --+ C ~+~ be a nonconstant holomorphic map, such that - f 0 z + f2 + . . . + f 2 ___ 0. Define ~ : N 2 X Z m+l --+ C by

~ ) : (w ,x ) H ( f ( w ) , x ) .

Then it is easily seen that the map ~ is a harmonic morphism in each variable separately. Following Proposition 4.3, the local solutions ¢: U C Z ~+ 1 __+ N2 to the equation ~(q~(x), x) = 0 are harmonic morphisms. The equation

+(u, , x) = 0

is invariant under the aI~+-action on Z ~+ l , i.e. if t~(w, x) = 0 then ~ ( w , Ax) = 0 for all A E I~: +. This means that every local solution can be extended to an ~ + - invariant solution ¢ : / ] C Z m+l --+ U 2 of the equation ~ ( ¢ ( x ) , x ) = 0. This will satisfy conditions (1) and (2), hence induce a harmonic morphism ¢: U --+ N 2 where U is the subset 7r((;) of H "~.

Every harmonic morphism from the hyperbolic 3-space H 3 to a Riemann surface has totally geodesic fibres. We now show how some of the well-known examples given in [4] and [5] fit into our notion.

E X A M P L E 5.1. Let ¢: a 4 --+ C be the ~+-invariant map given by

@: ('%'01Xl, X2, X3) ~-+ X2 + ix3

xo+ Xl

Then z --- q~ is a solution to ((z, z, 1, i), x) = 0, which means that ¢ defines a harmonic morphism ¢: H 3 --+ C.

E X A M P L E 5.2. L e t / 9 2 be the open unit disc in the complex plane on which we model the 2-dimensional hyperbolic space H 2 in the classical way. Let ~): Z 4 -+ H 2 be the ~,~+-invariant map given by

¢: (xo, x l, x2, x3) Xl -~ ix2

Then one easily checks that z = ~) is a solution to ((2z, 1 + z z, i (1 - z2), 0), z) = 0. This implies that q~ defines a harmonic morphism ¢: H 3 ~ H 2.


EXAMPLE 5.3. Let C = C U {00} be the Riemann sphere and let (7 be the open subset {x C Z4](xl ,x2, x3) 7 ~ 0} of 217/4. Then define the lI~+-invariant map ~;: (7 -+ ~ by

& (,0, ~I,,2,,3) Xl + ix2

• 3 + J q + 4 + 4

Then ~ = z is a solution of ((0,1 - z 2 , i ( l + z2) , - 2 z ) , x) = 0 and defines a harmonic morphism ~: H 3 - {(1,0, 0, 0)} -+ C:.

6. Local Solutions

In this section we give a method for constructing harmonic morphisms from odd- dimensional hyperbolic spaces H 2m+l with values in Riemann surfaces. We iden- tify the (2m + 2)-dimensional Minkowski space M 2m+2 with C m+l and use the complex notation

(Z0, Zl , . . . ,Zm+l) :---- (X0 q- i x l , . . . , x 2 m + iX2m+l).

Then it is easily checked that the conditions of harmonicity and horizontal conformality of a map q~: (7 -+ N 2 from an open subset (7 of Z 2"~+2 with values in a Riemann surface are: For every complex coordinate z: V --+ C on N 2

r ( z o q~) = - 2 [ O-z~ + 0-5~ + 4 ~-k-zk O g - 0, (3) k=l

~;(zoqS) = - 2 [ ~Zo J + 0--~o-J

+4EO(z°~) O(z o (b) ~=I - a g a - g - o.

(4)

PROPOSITION 6.1. Let (7 be an open subset o f Z 2m+2 and let (7* C C ~+1 be the image o f (] under the operation

• : z -- ( z O , ~ l , . . . , z m ) ~ z* := ( - (~o) , ~ l , . . . , z m ) ,

where cr(zo) := [(1 - i)zo + (1 + i)2o]/2 = xo + Xl. Further, let N 2 be a

Riemann surface and let F: N 2 × (7* --+ C be a holomorphic map. Then the map ~: N 2 X (7 --+ C given by

~ : ( ~ , ~ ) ~ v ( ~ , ~*)

is a harmonic morphism in each variable separately.


Proof. For each z E ~r the map ~ : N 2 --+ C is a holomorphic map between Riemann surfaces, hence a harmonic morphism.

Let w be an element of N 2, then differentiation of ~w with respect to z0 and 20 respectively yields:

O~o - ½(1- )U~' O~o ' ) ~ '

02~w . .202F 02~w ..2 02F

0 4 - ¼(1- ,) ~-~, 0~o~ _ ¼(1 + ,) b-~"

These equations and the fact that ~ o is holomorphic with respect to zl, z2 , . . . , Zm imply that ~ satisfies conditions (3) and (4). []

Let P1 , . - . , P~: C ~+l ---+ C be holomorphic homogeneous polynomials all of the same degree d and let f l , . . . , fn: N 2 --+ C be holomorphic maps. Define the map 6: N 2 x Z 2m+2 -+ C by

s=l

Proposition 6.1 implies that ~ is a harmonic morphism in each variable separately. The equation

is invariant under the R+-action on Z 2ra+2. Hence every local solution can be extended to an ~ +-invariant solution 6: U ~ C of ~ ((~(z), z) = 0. This induces a harmonic morphism ~b: U --+ C from the open subset U = 7r((]) of II 2m+1 .

COROLLARY 6.2. Let P, Q: c ~+1 --+ c be holomorphic homogeneous polynomials both of the same degree d and let U := {z E tt2"~+lIQ(z*) ¢ 0 or P(z*) ¢ 0}. Then the map (~: U --+ C U {c~} defined by

Q(~*) ¢: z ~ P(z*-~

is a harmonic morphism.

COROLLARY 6.3. Let P, Q, R: C m+l --+ C be holomorphic homogeneous polynomials all of the same degree d. Put

U~ := {z ~ Hz~+~IQ(z* ) ~ 0 or P(z*) ~ 0 or R(z*) ~ 0},


and

U2 := {z 6 H 2 ~ + I l Q ( z * ) e - 4 p ( z * ) R ( z * ) ¢_IRo}.

Let v/-: C - ]R 0 --+ C be the principal square root function r . e i¢ ~ v/~ • e i0/2,

where r E tR + and 0 E (-Tr, 7r). Then the maps 954-. U1 N U2 -+ C U {ec} given by

954-:Z ~+ - Q ( z * ) 4- ~ /Q(z*) 2 - 4 P ( z * ) R ( z * )

2P(~*)

are harmonic morphisms.

7. Complete Minimal Submanifolds

In this section we construct harmonic morphisms which are globally defined on the odd-dimensional hyperbolic spaces. They are used to construct complete minimal submanifolds of codimension 2 in H em+l which are not totally geodesic.

EXAMPLE 7.1. First we observe that the function

( zo , . . . , zm) ~ {r(zo) = (1 - i)zo + (1 + i)2o 2

= x o + x l

is strictly positive on the whole of Z 2m+2. Let P: C m --+ C be a holomorphic homogeneous polynomial of degree d > 1. Then define two maps ¢1: Z 2m+2 --+ C and 952:z2m+2 ---+ C U {(x3} by

P(zl,...,~) ~;~:(~o,...,~) ~ ~(zo)~ (5)

- P ( z l , . . . , Z m )

q;:: (~o,..., z~) ~ P ( ~ I , . . . , ~ ) + ~(~o) d (6)

The maps ¢1 and @2 axe ~+-invariant and both satisfy equations (3) and (4) so they induce globally defined harmonic morphisms 951://2m+l __+ C and 952://2m+l __+ c u {oo}.

As already mentioned in the Introduction, all regular fibres of a harmonic morphism 95: M --+ N 2, with values in a surface, are minimal submanifolds of M of codimension 2. We will now use this fact to construct such manifolds in/ /2m+l.

EXAMPLE 7.2. Let the harmonic morphism 95://2~+1 __+ C be given by

¢: (z0,..., z ~ ) ~ zf + 4 + . - -+ 4 a(zo)d (7)


The crit ical set o f ¢ is g iven by

C¢ : {(zo, z l , . . . , Z m ) E H2m+l[z l = z2 . . . . : Zm = 0 ) ,

SO 0 E C is the on ly critical value o f ¢. This means that for each non-ze ro ~ E C

the fibre

d ¢ - 1 ( ~ ) = {(Zo, Zl, . . . ,Zm) e H2m+tlz~ + z~ + ' " + zm = c~'a(zo) d)

is a min ima l subman i fo ld o f the hyperbo l ic space H 2m+l . It is obv ious ly comple te

and if d > 1 it does not lie on any linear 2m -d i m e ns iona l subspace o f the M i n k o w s k i space M 2m+2 and cannot therefore be total ly geodesic.

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minimal submanifolds of hyperbolic spaces via harmonic morphisms

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