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ni arm
ym
Thet,1tio0,
md
1C
n. jt
h
7Ps
ar
ono
ms
1C
is etr 
de
Technology
of
Institute
usetts
PBa. jrd
Massach
PITMAN BOOKS LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050 Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto
© P Baird 1983 First published 1983 AMS Subject Classifications: (main) 58E20, 53C40 (subsidiary) 58F05, 70E15
Library of Congress Cataloging in Publication Data Baird, P. (Paul) Harmonic maps with symmetry, harmonic morphisms and deformations of metrics. (Research notes in mathematics; 87) Includes bibliographical references and index. 1. Harmonic maps. 2. Submanifolds. I. Title. II. Series. QA614.73.B34 1983 514'.74 838186 ISBN 0273086030
British Library Cataloguing in Publication Data Baird, P. Harmonic maps with symmetry, harmonic morphisms and deformations of metrics.(Research notes in mathematics; 87) 1. Harmonic maps 1. Title II. Series 514'.74 QA614.73 ISBN 0273086030
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers.
Reproduced and printed by photolithography in Great Britain by Biddies Ltd, Guildford
Preface
We consider the basic problem of harmonic maps, that of finding a harmonic representative for a given homotopy class of maps between two Riemannian manifolds. Given that we are unable to solve this problem for fixed metrics, we can ask whether there exist metrics with respect to which there exists a harmonic representative this is known as the 'rendering problem'. In particular, we study these two problems for maps between spheres. We do this by considering maps where 'reduction occurs', that is, because the map possesses certain symmetries (equivariant), the problem of harmonicity reduces to solving a certain secondorder nonlinear ordinary differential equation (the reduction equation). The origin of this method is the Thesis of R.T. Smith (3972), in which certain harmonic maps between Euclidean spheres are constructed. The symmetry which we make use of is that the map should preserve families of parallel hypersurfaces with constant mean curvature. It should be noted that this seems to be a more natural symmetry in the context of harmonic maps than the more commonly exploited symmetry of 'equivariance with respect to group actions'. We prove a reduction theorem for harmonic maps between space forms, and provide many examples of maps satisfying the conditions of the theorem. In some instances the reduction equation is easily soluble, on the other hand we find examples where we have little idea about appropriate solutions to the corresponding reduction equation.
Using the stressenergy tensor associated to harmonic maps we study harmonic morphisms, proving a theorem characterizing those harmonic morphisms with minimal fibres. We then consider harmonic morphisms defined by homogeneous polynomials, relating such maps to the construction of maps between spheres where reduction occurs. By allowing deformations of the metrics, first for harmonic morphisms and then for maps where reduction occurs, we are able to solve the rendering problem for all
classes of the homotopy groups 7r (Sn) = Z for all n. n Recently it has become apparent that harmonic maps have a significant role to play
in certain problems of theoretical physics. A particular example is the description of solitons in terms of harmonic maps from S2 into the complex projective space CPn . It is to be hoped that this work will lead to a greater understanding of solutions with symmetry of some of the variational problems of physics. I would like to express my most sincere thanks to Jim Eells who has provided continued support and encouragement during the preparation of this work. I would also like to thank L. Lemaire, J. Rawnsley, H. Sealey and J.C. Wood for many helpful conversations, the Science Research Council for their financial support, and the Institut des Hautes Etudes Scientifiques and the University of Bonn for their hospitality and support during 1981.
I am also especially indebted to my parents who have provided encouragement throughout.
Paul Baird
Contents
p.
INTRODUCTION
1
1.
FIRST CONSTRUCTIONS
7
1.1
The Laplacian on the sphere
1.2
1.4
Harmonic maps into spheres Joins of spheres and Smith's construction Outline of the solution of Smith's equation
1.5
Hyperbolic space
19
1.6
22
1.7
Polar coordinates on hyperbolic space and an analogous construction to Smith's Solving the equation for hyperbolic spaces
2.
ISOPARAMETRIC FUNCTIONS
27
2.1 2.2
Definition of isoparametric function Properties of isoparametric functions and Munzner's classification
2.3 2.4
theorem Examples of isoparametric functions Generalizing the notion of isoparametric families of hypersurfaces
3.
THE STRESSENERGY TENSOR
42
3.1
Derivation of the stressenergy tensor
42
1.3
3.2
7
9 11 16
24
27
29
34 39
46
3.3
The eigenvalue decomposition of the stressenergy tensor
4.
EQUIVARIANT THEORY
4.1
Maps which are equivariant with respect to isoparametric functions 49
47
49
4.2
Generalized equivariant maps between Riemannian manifolds
65
5.
EXAMPLES OF EQUIVARIANT MAPS
69
5.1 5.2 5.3
Maps from Euclidean space to the sphere Maps from hyperbolic space to the sphere Maps from sphere to sphere
69
6.
ON CERTAIN ORDINARY DIFFERENTIAL EQUATIONS OF THE PENDULUM TYPE
76
80
94
6.1 6.2 6.3
Existence of solutions Asymptotic estimates Smoothness of certain equivariant harmonic maps
7.
THE GENERAL THEORY OF HARMONIC MORPHISMS
122
7.1
General theory
122
7.2 7.3
Examples and nonexamples of harmonic morphisms
125
94 111
116
Maps 0 : (M,g)0(N,h) where 0 * h has two distinct nonzero eigenvalues
133
8.
HARMONIC MORPHISMS DEFINED BY HOMOGENEOUS POLYNOMIALS
8.1 8.2 8.3
Properties of harmonic polynomial morphisms 135 Some examples 139 Harmonic morphisms defined by homogeneous polynomials of degree bigger than two 142 Harmonic polynomial morphisms and equivariant maps between spheres
135
8.4
146
9.
DEFORMATIONS OF METRICS
153
9.1 9.2 9.3
Deformations of the metric for harmonic morphisms Examples Deformations of metrics for equivariant maps
153
160 162
9.4
Examples REFERENCES INDEX OF DEFINITIONS
172
177
180
Introduction
Let (M, g) denote the Riemannian manifold M together with its metric g.
Let 0 be
a map between Riemannian manifolds:
0: (M, g)
(N, h)
(all manifolds, metrics and maps will be assumed smooth unless otherwise stated). Then the derivative of 0, do , is a section of the bundle of 1forms on M with values in the pullback bundle Qf 1 TN: dO
c W (T*M ® 01 TN).
The bundle T*M®O1 TNis a Riemannian vector bundle and has a LeviCivita connection V acting on sections:
V: W(T*M ®01 TN)p_W(®2 T*M ®
01 TN).
Call v d 0 E e9 (02 T* M ® p11 TN) the 2nd fundamental form of the map 0 . Then we say that 0 is harmonic if trace V do = E vdp (X ,X .) is zero at each i 1 g point of M, where (Xi) I< i < dim M is a local orthonormal frame field for M. Given a map 0 as above; there are two basic problems of harmonic maps: (i) letting [0 ] denote the homotopy class of 0 , then does there exist a harmonic
representative 0 E [0 ] such that 0: (M,g)w (N,h) is harmonic? ( ii) Do there exist metrics g and h on M and N respectively, such that there h) harmonic? exists a representative 0 E [0 ] with 0 : (M,g) The first of these problems has been considered throughout the history of elliptic analysis and differential geometry. The first existence theorem was proved by Eells and Sampson in [ 12 ]. The second problem is known as the rendering problem. This has been considered by Lemaire [28 ] in the case when the domain is a surface. One of the main objectives of this work is to study these two problems for maps 0: Sm PP.. Sn between spheres, where the homotopy classes are represented by the groups
Tr 1
Up until about 1970, the only harmonic maps 0: Sm . Sn, m > n, that were known were a few homogeneous polynomial maps (defined by eigenfunctions of the Laplacian on Sm) :
(a) the identity maps In : Sn  Sn for all n > 0 ; n : Stn1 Sn , for n = 2, 4 and 8; (b) the Hopf maps H k, I where z e c , I z I2 = 1 and k is any : z Z (c) the maps Gk : S1
S
integer. In his Thesis in 1972, Smith [33 J gave a method whereby, given two such homogeneous polynomial maps, and provided certain "damping conditions" are satisfied, one can construct another harmonic map between spheres. Smith's method was to construct a 1parameter family of maps all homotopic to the join of the two polynomial maps. This family is parametrized by a function a from the interval to itself. Smith then exploited the symmetry of the so constructed map OM to reduce the problem of whether 0a is harmonic or not to solving a 2nd order nonlinear ordinary differential equation in a . This procedure of reducing the problem of harmonicity of a certain map to solving an ordinary differential equation we shall simply call reduction . The corresponding differential equation we shall call the reduction equation. Smith's equation has a simple physical interpretation  that of a pendulum moving under the influence of a variable gravity force with variable damping. The above damping conditions are sufficient conditions for this equation to have a solution. We place Smith's method into a more general setting by viewing his maps as sending "wavefronts to wavefronts"in the sense of geometric optics. That is, we have a commutative diagram of the form:
M
Oa
N
f
g If
a
g
where f, g are real valued functions on M,N respectively, with values in intervals I f9 19 2
respectively. , and a: I f  19 is a smooth function. The appropriate class
of functions to which f and g must belong in order to provide a good theory turns out to be the class of isoparametric functions as defined by Cartan in 1938 [5 ]. These objects provide a rich and beautiful geometry on spheres, and it is indeed pleasing to find that they should be related to constructing harmonic maps between spheres. After Cartan's impressive study of isoparametric functions in papers dated 1938 1940 [5,6,7,8 ], the subject lay dormant until Nomizu's paper of about 1970 [32 ], surveying Cartan's work and giving some open problems. That revived interest in the subject, and Mummer proved an important classification theorem [30 ]. Also many new examples of isoparametric functions on spheres were found, see [32,34,40,33,17] In particular, examples of isoparametric functions on spheres whose level surfaces are nonhomogeneous were found by Ozeki and Takeuchi [33 ] and Ferus, Karcher and Munzner [ 17 ]. These examples are very important from our point of view, since we make essential use of them in Theorem 5.3.8, Example 8.2.2 and in Section 8.4. This demonstrates that isoparametric functions are more natural for our purposes than families of homogeneous submanifolds of space forms (from which point of view one could conceivably derive our theory). R. Wood was the first person to make a connection between isoparametric functions and harmonic maps [43 ]. One of the essential features he observed was that such functions satisfy an eikonal equation of the form
Idf(x)I2
clxl2p2
=
for some constant c, where p is an integer, p > 2. This remarkable fact makes the connection with geometric optics even more striking. Isoparametric functions as studied by Cartan are defined on space forms  that is, complete connected manifolds of constant curvature. In order to consider the construction of harmonic maps between more general Riemannian manifolds, we make what we consider to be a suitable definition of an isoparametric function on an arbitrary Riemannian manifold in Section 2.4. In Chapter 4 we develop the general framework, and define the necessary conditions on 0 in order that reduction occurs. Maps 0 satisfying these conditions we call equivariant. We prove our main theorem (Theorem 4.1.8) which is a reduction theorem for harmonic maps between space forms. 3
As a consequence of the above generalization, we can construct explicit harmonic maps from Euclidean spaces to spheres and from hyperbolic spaces to spheres  this is done in Sections 5.1 and 5.2. In Section 5.3 we find a very large number of classes n (Sn) where reduction occurs. However, many of the corresponding m
reduction equations have a qualitatively different nature to those of Smith, and we are
unable at present to solve many of them. One important class of equations does have similar properties to those of Smith, and the solutions yield some new and interesting harmonic maps between Euclidean spheres of the same dimension. This is proved in
Theorem 5.3.8. Two of the most significant examples in Smith's Thesis are
(i) the construction of harmonic representatives for all classes of Tn (Sn) for all n < 7, and which are parametrized by ( ii) the consideration of the classes of IT3(S2) their Hopf invariant (see [26 ]). The important point in example (i) is that the condition n < 7 is a consequence of the damping conditions. Thus Smith's method fails to give harmonic representatives for the classes of 11 n(Sn) with n > 7. In example (ii) the classes of TT 3(S2) are parametrised by their Hopf invariant d e Z. When d is the square of an integer, d = k2 for k e Z ; Smith gives a harmonic representative of this class. However, when d = k 1, k,1 c Z, k,1 0 and k 1; Smith provides a representative of the class where reduction occurs, but demonstrates that the corresponding reduction equation does not have a solution.
With the above two examples in mind we consider deformations of the metrics on space forms (Chapter 9)  this fits quite naturally into our general framework. We thus come to one of the main consequences of the reduction theorem. This is the solution of the rendering problem for all classes of nn(Sn) for all n (Theorem
9.4.5). The deformed spheres are familiar ellipsoids whose eccentricities depend only on a and the degree. Even allowing various deformations of metrics, we are still at present unable to solve the rendering problem for i 3(S2) . In Chapter 7 we undertake a general study of harmonic morphisms  maps between Riemannian manifolds which pull back germs of harmonic functions to germs of harmonic functions. We prove a theorem characterizing those harmonic morphisms for 4
which the fibres are minimal submanifolds. This generalizes a result of Eells and Sampson 1121 stating that every Riemannian submersion which is harmonic has minimal fibres. Harmonic morphisms were first considered in detail by Fuglede [ 19 ], and we make use of his ideas in Chapter 8 when we consider harmonic morphisms defined by homogeneous polynomials. Here we find an interesting connection with isoparametric functions, and we prove a theorem which associates to a certain class of harmonic polynomial morphisms an interesting harmonic Riemannian submersion. By using this theorem we can construct more maps between spheres where reduction occurs. One of the important tools used in Chapters 7, 8 and 9 is the stressenergy tensor a divergence free symmetric 2tensor field on M. Such objects are wellknown and important in relativity theory, where they in some sense model the matter distribution in a spacetime model. For if (Sil ) 1 < i,j for the Euclidean metric on JR n
< d(jo0), j o 0 > = 0. We differentiate this formula at x E M. Let (X a ) 1
M given by
u1vl + u2v2 + ... + um vm
M
for all u = (uI, ... , um) , v = (v1, ... , Vm) r IRm. The space Mm is called mdimensional Minkowski space. Define (m1)dimensional hyperbolic space, or
the (m1)dimensional pseudosphere, to be the space H
m1
={U
Mm
< u,u >M = 1)
;
,
together with the induced metric. HM1, The pseudosphere is a "spacelike"hypersurface (if v is tangent to then < v, v >M > 0), whose unique normal vector field r is "timelike "(< n , n >M < 0) in the terminology of relativity theory. There are many analogies between the Euclidean sphere Sm1 and the pseudosphere Hm1, especially concerning harmonic maps. In particular, the usual stereographic projection sending the sphere less a
point to Euclidean space, can be modified to give the wellknown isometry between M1 m1
and (B m1, < , >) , where B m1 is the open ball of radius 2 in R M1 dx. ® dxi/ (1  4 £ x 2 2 , where (x.) are the and the metric < , > = E H
standard coordinates on Euclidean space. Let is H m1
Lemma 1.5.1 H
M1
where r/c^r
A M
m
M in be the inclusion map. If
f: Mm IR is a smooth function, then m Mm
(foi) _
.2 f+
c
+(m1)
f2
2r
Ff
)oi
denotes differentiation in the normal direction to 2 `2
ex
2 +
8x2
+
,,.
+
(1.5.1)
,
C71
8
2
Hm1
and
is the indefinite Laplacian.
rx m 19
Remark 1.5.2 The unit timelike normal to the pseudosphere at x is x itself. Thus, ^c (x) = x1Fx1() + x2
^c
.) r
(x) + ...
Fx2
+x
m m Fxm
(x)
where (x1. ... , xm) are standard coordinates on JR m Proof (of Lemma 1.5.1) : Recall equation 1.2.1 for the 2nd fundamental form of the composition :)f two maps:
vd(f o i) = df(Vdi) + Vdf(di,di)

this makes sense with M m having an indefinite metric (V is the LeviCivita connection with respect to the indefinite metric on the bundle T * H ml i l T M m H ml) . By taking the trace of this formula with respect to the Riemannian metric on H m 1, we get A
H ml
(foi) = df(A
H m1
+ trace Vdf(di,di).
i)
f
But, (A
Mm
f) o i = trace V df(di,di) ` m
=
2
1
2
22_
+
2
+
,,,
+
x2
(1.5.3,
F2
is the indefinite Lapla
Fx2
m
f)oi = A HM1 (foi) 
2 oi 8 2
`
Now
Hm1 i
trace V d i m1 Mm Vdi(X ) di (Xk) 1 k
k 20
oi
Equations (1.5.2) and (1.5.3) give
cian on M m. Mm
2
2 Fx
A
2
Fr2
where A M
(A
(1.5.2)
dfCHm
i
.
(1.5.4)
where we evaluate at a point x ,
Hm1
with Xk = yk (0), yk (0) = x, yk(u) geo
desic in Hk = 1, ... , m1. Thus Thus
yk(0) = 8/aij
pH mi
pH m1
i=
11
yk (0). But yk (0) _
0
i =(m 1)
If f is a pseudoharmonic polynomial (i.e. 6M Corollary 1.5.3 degree k on R m, then pHm1
so foi
m
f = 0) of
(foi) = k(k+m2) foi;
is an eigenfunction of
AHm1
.
Let (M,g) be a Riemannian manifold and 0: M be the inclusion map, and write 4 = i o 0 . Lemma 1.5.4
Hm1
a map.
Leti:Hm1y
Mm
The map 0 is harmonic if and only if
p = 2 e (fi) D
,
where a (1) = E 2 .
Then f = F I S2n+1 is isoparametric of degree 4 on S2n+1 Define is : S1 X s1,2  S2n+1, where S n+1, 2 is the Stiefel manifold of orthonormal 2frames in Rn+l, by i
s (e i0 , (x, y)) = e ig (toss x + i sins Y)
where s E [0,11/4 ]. Then is is an immersion which double covers the level hypersurface M S = f 1 (sin2 2 s). The hypersurface is obtained by the identification ( 0 , (x,y) )  (0 + 11, (x,Y) ).
The principal curvatures of M. are cots, cot(s  11/4) , cot (s  11/2), cot(s  311/4) with multiplicities n1, 1, n1,1 respectively. There are two focal varieties, one at s = 0, which corresponds to the set { e i0.x i = S1 XSn/SO, and one at s = 11/4, which corresponds to the set {ei0(x + iy)/22' It , i.e. the set{(x+iy)/22? _ Sn+122 . Let us study the Riemannian geometry of Ms in more detail. Define Sn+1 2 to be the analytic submanifold of Rn+le Rn+1 given by Ss
+1 2 =
{ (x,y) E Rn+le Rn+1; Ix 12 = cos2s, ly 12 = sings and <x,y> = 0} = {(toss x, sins y) C Ra+1® Rn+1 (x y) , E Sn+12 2 } . Let a1, ....en+1 be the orthonormal basis for Rn+l
such that ei = (0,... ,0,1,0,... ,0), with the 1 in the i'th place. Choose pESn+1,2 to be p = (cosse1,sinse2). Consider the following curves in Rn+1
Ai(u) =cos(u/sins) e2 + sin(u/sins) ei, i =3,...,n+1 36
,
(2.3.3)
and define the curves through p in Sts
1,2
yi(u) = (coss yi(u), sins e2) (u)
p(u)
= (cosset, sins Xi(u)), i = 3, ...
, n+1
= (coss(cos u e1 + sine e2) , sins (sinuel + coss e2)) .
(2.3.4)
The tangent space to Sn+1,2 at p is given by
TpSn+1,2 = { (v,w)p; < v,w> = 0 = <w,y> and + <x,w> = 0 where p = (x,y))
,
and the vectors
y'(0) = (ei, 0)p x'i (0) = (0,ei)p
P'(0)
,
i = 3, ...
, n+1
= (coss e2, sins el) ,
(2.3.5)
form an orthonormal basis at p for TpSn+1,2 Lemma 2.3.6
The curves (2.3.4) are all geodesic in S n+1,2 at p  that is,
E n+1 e )Rn+1 /.i'(0) )p is perpendicular to TpSn+1,2' where p (u) is one of (VQ,(0)
the curves (2.3.4). Proof A curve /3(u), (2(0) = p, in Sn+t,2 is geodesic at p if and only if p"(0) is perpendicular to TSn+1, p For example, p"(0) = (coss (e 1 ), sins ( e 2) ). 2 Clearly the scalar product of p "(0) with the vectors of (2.3.5) is zero. Similarly for the other curves of (2.3.4). FI We study M , the double cover of M s , which locally can be described as the set s of points ei6 (coss x + isins y), s E (O, 1T/4), (x, y) E Sn+1,2 and B E [0,211). Fix g , then S n+1, 2 is embedded in (rn+l as the manifold S n+1, 2 . Fix coss x + isins y, then as B varies, we trace out a great circle of S2n+1. How does this
circle intersect the Ssn+1, 2 37
Fix p = e
i00
(toss x0 + isins y0) E M s . Consider the two curves
y (u) = e
i9 (u)
6(u) = e 1
00
(toss x0 + isins y0)
(coss x(u) + isins y(u)) ,
(2.3.6)
both of which are contained in Ms, where x(0) = x0,y(0) = y0, 0(0) = 90 and (x(u), y(u)) E Sn+1 2 for all u. These are both curves through p, and without loss of generality 0 '(0) = 1. Then
y (0) = i9 =
(0) ai
00
(coss x0 + isins y0)
ei 00(sins y0 + icoss x0)
and
6'(0) = ei00(coss x'(0) + isins y'(0))
.
The Riemannian scalar product
dx,
where S0 C' (02 T *M) , called the stressenergy tensor of 0 , is given by
2
S L + 2 g L, and < , > is the metric induced on 02 T * M from g (by " L we g
0
g
mean the section of p T* M, which is given in components by (2g L) ab gabdxa dxb with respect gacgbd (summing over repeated indices), where g = cL 2gcd to a local coordinate system (xa ) on M). Proof Suppose g has a local representation in the form g = gabdxadxb (summing over repeated indices) with respect to local coordinates (xa) on M. Then
dI u 2L du u = 0M 2gab gab dx + ./M a J
L
2(dx) 0gab
a gab
The volume element dx can be written in the form dx = (det g)Zdx1 A ... A dxm. Then
i (dx)
2(detg)2 (cofactor of gab) dxl A ... ndxm
2 gab
2(det g)1 (cofactor of gab) (detg)2 dxl A ... A dxm g
Corollary 3.1.3
ab If
dx.
0 and L are as in Example 3. 1. 1, then
S0 = e(0) g  0 * h Proof Let (x a, ya) be a local coordinate system on E = M x N. These induce coordinates (xa,yx,ya) on J1(E) with respect to which L(xa,y , ya ) gabyayb hap, where g = gabdxa dxb and h = ha#dyady are local representatives of the metrics g,h respectively (summing over repeated indices). Then
2gab But gij
gj k
=
(
ag1l
ab
) ya Yj#ha
= b k , so
0 = "gab
gjk
+ g t)
gjk
gab
43
Whence gll
_
gab

g
it
`glk
g
jk
gab
gil sa ab k
1
jk
g
gia gjb Thus 2L/P gab
 yaayfibhaR
Proposition 3.1.4 M
Or in coordinate free notation 2 L(0)
0 *h.
g
LJ
If X is a smooth vector field on M, then
dx  2'r V* S0(X)dx = 0 M
where V * S0 denotes the divergence of S0 ; V * S0 (X) = E VX S0 (Xi ,X) , where (X. ) i is an orthonormaI frame field.
Proof Let to be the family of diffeomorphisms associated to the vector field X, then for each u I = '
1M
Ldx
fthu(M)Ldx
f ?u*(Ldx). M
Therefore
fM(Ldx  thu *(Ldx)) = 0, and so f 2X(Ldx) =0, M
where YX denotes Lie derivation with respect to X. Also
X(Ldx) _