c

Harmonie Maps

© Christopher Kumàr Anand Department of Mathematics and Statistics

McGill University, Montréal

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of

Masters of Science

24 April 1990

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Contents

0 Introduction. 1

1 Preliminaries. 8 1.1 Definitions. 8 1.2 (Counter )example. 10 1.3 Tension Field. . . . 13 1.4 Geodesics. 14 1.5 Surfaces ....... 16

2 Nonpositive Riemannian Curvature. 22

3 The case dim M = 2. 26 3.1 Sobolev Spaces. . ........... 27 3.2 Stretching the Sobolev Inequality.. . 29 3.3 The Application.

••••••••• 1 • 34

4 The case dimM = 2 and 7r2(N) = o. 37 4.1 Morse Theory. .. 38 4.2 Perturbation. . . . . ......... 41

5 Conclusion. 52

1

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Abstract

After a brief int.roduction, we consider three main results in the existence theory of harmonie maps between manifolds. The first is the heat-equation proof of EeUs and Sampson, which says that minimal harmonie maps of compact manifolds into compact manifolds with nonpositive curvature always exist. The next two results show they exist among maps of compact Riemann surfaces into compact manifolds, N, with 'Tr2(N) = O. One proof uses the induced 'TrI-action of Schoen and Yau; t,he other a perturbation of the action due to Sacks and Uhlenbeck. As required, wc aiso develop sorne of the regularity theory, especially that for surfaces.

Resumé

Après une brève introduction, nous considérons trois résultats importants dans la théorie des applications d'une variété Riemanienne dans une autre. Premièrement, nous expliquons la méthode d'équation de chaleur d'EeUs et Sampson pour dps applications d'une variété compacte dans une autre, aussi compacte, avec courbure nonpositive. Les deux autres résultats concernent le cas d'une surface de Riemann dans un espace, N, compact avec 'Tr2(N) = O. L'un utilise l'action induite sur 'TrI, de Schoen et Yau; l'autre, une perturbation de l'action de Sacks et Uhlc1l1)('ck. Nous développons aussi un peu la théorie de régularité, à mesure qu'elle est requise, surtout pour les surfaces.

Chapter O.

Introduction.

In 1930, Douglas and Rado showed that a simple closed curve in Rn whieh hounds

a dise of fini te area bounds a disk of minimal area. In 1948, Morrey extended this

result to homogenous regular manifolds, which includes closed manifolds. This ean

he ealled the first modern result in the existence and regularity theory of harmonie

maps between manifolds. In faet, it was Morrey, who in Multiple IntegraIs in the

Calculus of Variations laid much of the groundwork analysis for the field.

Harmonie maps are a generalisation of both harmonie functions on eomplex

domains and geodesies in manifolds. Unfortunately, the theory of harmonie maps

is much more complieated than the eorresponding theory for harmonie funetions.

Given a harmonie function on sorne open domain in R2, it is elementary to find

holomorphie funetions, loeally, whose real part is given by the harmonie funetion.

That holomorphie funetions (and henee harmonie functions) are smooth follows

from fil'st principles; that harmonie maps are regular is a major stumbling bloek in

highcr dimensions. Harmonie maps are defined to he the extremals of an energy

( .. functional EU) = lM 1 df 12 dXM. Sueh a funetional is not new to analysis,

1

.... and in other settings, the calcul us of variations is usually brought to bear. The

Euler-Lagrange equations for this functional ean be determined, and will he writtcn

TU) = O. The form of T, in general, a3 well in several examples, will be giV('n in

the main text. We should mention that the notation for T is not universal. SOIll<>

authors use 6, sinee for maps into fiat manifolds, r is just the Laplacian. This

gives us the connection with harmonie functions, which are harmonie maps into fiat

manifolds. Unfortunately, other aut,hors use 6, loeally, to reprcsent the Laplace­

Beltrami operator acting on eoordinate functions of a trivialisation.

Although, in general, the form of ris not so pretty, the Euler-Lagrange formula­

tion is indispensible in its role of transforming the harmonie map prohlcm (i. c. the

question of the existence of harmonie maps) into a problem in the thCOly of Partial

DifferentiaI Equations (PDE theory). PDE theory can, for our purposes, be broken

into two bodies of results: the existence theory and the regularity theory. Regu­

larity essentially asserts that an harmonie maps in sorne larger space (the spacC' of

continuous maps, or LP-maps, for example) are actually regular, z.c. thcy bclong to

sorne nicer subspace of maps, such as the smooth ones. Regularity rcsults arc local

results- they do not, a priori, depend on the topology of the manifolds They do

depend on the local form of rU) = 0, in terms of fixed coordinate systems. Locally,

TU) = 0, is just an elliptic system of PDE's. AIthough the rcgularity theOl'y is

quite well-developped, and immensely important, we will not explore it in detai! in

this thesis.

2

( Instead, we will concentrate on the existence theory, and several important in-

novations therein. There are two basic existence problems for harmonie maps:

1. Given Riemann manifolds M and N, and a class of maps M ~ N (for ex-

ample, aIl maps 10motopic to a fixed map, an maps, or aIl maps inducing a

given map of 'Trl Uvl) into 'Trl(N)) does E take on its minimum on this class of

mups, z. e. does this class have a minimal harmonie representative.

2. Given M and N, and a class of maps, does E have critieal points on this class,

i. e. does it have a harmonie representative.

This thesis will be logically divided into four chapters. In the first ehapter,

we will define the functional E and give an exarnple of a class of maps of spheres

which do not contain a minimising harmonie map, because E does not attain its

minimum. This ex ample will give us reason to work out the energy integral in a

fairly straight-forward instance. Although, in general, we will not be caleulating

EU) and 100 king directly for minima, this ex ample should give the reader sorne

intuition as to the sort of obstructions to finding minima. Having shown that

the harmonie map problem may have a negative answer, we proceed to the case of

geodesics, where the most naive variational methods show it has a positive solution.

To see that harmonie maps from an interval into a manifold are geodesies, we will

show that they are solutions to the same differential equations, i.e. the geodesic

equations and the Euler-Lagrange equations for E are the same. This, of course,

( llwans that we must know the form of these equations. In section 1.3, we write

3

1 ]

J

1 1

them loeally as a vector field whieh plays the role of the gradient of E. Beeause of

its role in the Heat Equation proof of the existence of harmonie maps. we eaU this

field the tension field eorresponding to E.

Unlike the case of (initial manifold) dimension one (i. e. geodesies). we cannot

hope to apply naive variational methods to the dimension-two case. In section 1.5,

we will look at the relationship between the Plateau problem and the harmonie

map problem. In fact, the two problems are linked by eonformality, ,. e. conformai

immersions are harmonie if and only if they are minimal surfaces. Having estab­

lished this, we outline conditions guaranteeing that a harmonie map is a conformai

branehed minimal immersion.

Chapter 1 serves to introduce the concept of Energy, and develop sorne of the

basic machinery, but it also hints at one of the underlying diffieulties in proving

results about harmonie maps: the dimensional problem. The naive method which

works for geodesics will not work for surfaces, and the tricks used on surfaces will

not generalise to three dimensions. Unlike many problems in mathematics, the

harmonie map problem is very sensitive to dimensio:1.

The remaining three chapters discuss three significant results in the existence

theory. Chapter 2 deals with the Eells and Sampson's proof that energy-minimising

harmonie maps exist in aH homotopy classes of maps between two compact mani­

folds if the second has Riemannian curvature tensor everywhere nonpositive. They

find these minimising representatives as the limit of a heat diffusion. This proce-

4

( dure was inspired by an earlier proof of the existence of harmonic forms by Milgram

and Rosenbloom (see [Mil-Ros]). Another proof was later given by Uhlenbeck (in

[UhU]) using the well-known perturbation method, which we will describe in sec-

tion 4.1. The heat-equation proof imitates the classical-analytic solution of the heat

equation, depending on cleverly derived estimates.

In chapter 3, we introduce the Sobolev spaces of maps, and the Fundamental

Theorems of Sobolev Theory. Sobolev spaces are spaces of maps differentiable

in the distribution sense in an LP space. Given a continuous map f : M ~ N

of manifolds, we can define a map 'Trl(f) : 'Trl(M) ~ 1f'1(N), in a natural way,

sending loops (1 : SI -+ M into f 0 (1 : SI -+ N. Section 3.2 con tains a proof that

f E L~(M, N)-a map (really a class of maps equivalent in the L~-norm) which

is not necessarily continuous-also induces such an action. We apply this result,

important in itself, in the following section, to show that in the class of Li-maps of

a surface into a compact manifold, inducing the same 'TrI -action up to conjugation,

E attains its minimum. Such a map is harmonic, and certainly energy-minimising

in its homotopy class.

The last chapter deals with a different approach, which adapts an old framework

(Morse thcory) ta a broader field of application. In section 4.1, we give the briefest

trcatment of Morse Theory, as generalised by Palais and Smale. Basically, Morse

theory attempts ta get topological information about M by counting the critical

if points of a smooth function on M with isolated critical values, satisfying a conver-.

5

gence condition (referred to as condition (C». In the case of geodesics, E satisfies

this condition on the Sobolev space L~(R, N), which is a (infinite-dimensional) man-

ifold. This is, however, not true in general, which is where Uhlenbeck's innovation

is involved. Instead of E, she considers a functional close to E, È say, and shows

that Ë does satisfy (C) on 9. good manifold. In section 4.2, we give the perturba­

tion argument Sacks and Uhlenbeck used to show that if M and N are compact

Riemannian manifolds, dim M = 2 and 7r2(N) = 0, then an hornotopy classes of

maps have energy-minimising harmonie representatives. The idea of the proof is to

consider functionals

EOI(f) = fM(l+ 1 df 12t dXM

for a ~ 1. Note that El = E + 1 and EaU) > EU) + 1 for aH J. When fi' > 1,

Ea is a good functional on the (Finsler) manifold L~OI(M,N), whose extrema are

smooth. Ell'-minimising maps exist, for Q' > l. Considering finite-energy maps,

we investigate what type of convergence we can expect from a sequence Jo of Eo-

minimising maps, for sorne sequence of a's descending ta 1. What wc filld is that,

in neighbourhoods (of M) where 1 dfOi 1 is bounded uniformly, the convergence is

ct, but wh en the derivatives 'blow up' at a point, a sphere 'bubbles off' as Q' -- 1,

and the image of M in N contains the image of a harmonie map, plus this sphere,

in the sense that there exists an J : M -+ N harmonie and a fI : S2 -. N, such that,

6

( and

E(f) + E( (7) ~ liminf Ecr(fcr) - 1. or .... l

Although in a different context, the nonconvergence of tl-e maps of Riemann spheres,

described in section 1.2, is a prototype for this type of bubbling off. Since the

nonconvergence of ICi' as a -+ 1 manifests itself in such bubbling off, we should ask

what happens when there are no (homotopically) nontrivial spheres to be bubbled

off. The condition 7r2(N) = 0 turns out to insure that the convergence is good, and

hence that the perturbation method finds minimising harmonie maps in the correct

homotopy class. With this result, we close chapter 4.

Throughout this thesis, 1 have tried to make the material understandable ta

someone with no knowlege of advanced or specialised tapies beyond 'elementary'

differential gt:ometry and manifold theory. Where it has been necessary to assume

more advanced results-and tl:is has happened too frequently-I have tried to give

references which will be of help to the novice. 1 have also included a large bibli-

ography, in case these references are not sufficient. Finally, 1 would like to thank

Professor Jacques Hurtubise for his guidance and patience.

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7

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Chapter 1

Preliminaries.

1.1 Definitions

In the following M and N will always be smooth, Riemannian manifolds, M com-

pact, m = dimM, n = di;aN, with metrics 9 and h respectively. T(X) will always

denote the tangent space of a manifold. Given f : M -+ N and F a vector bundle

over N, 1* F will denote the pull-back bundle over M. Throughout this thesis, ~

means homotopie and ~ means isomorphic.

Following [Eells-Samp], we define eU), for f : M -+ N by

e(f)(x) = ~ 1 df(x) 12

where 1 df(x) 1 is the Hilbert-Schmidt norm of df(x) E T;(M) 0 TJ(x)(N). In

coordinates,

Th function e(f) is sometimes called the energy density of f. Intcgrating, wc

obtain the energy of f,

8

a function on COO(M,N), whose definition is motivated by the concept of kinetic

energy in mechanics. The volume form on Mis

in coordinates. We will assume it is normalised, i.e. that fMdxM = 1. Harmonie

maps are defined to be the extremals of energy. We would especially like to find

minima of this functional. Of course, one might expect that eU) can be defined

invariantly. In fact, eU) =< g, f* h >, wht!re the inner produet is that for 2-

covariant tensors of TM, induced by the Riemannian metric.

Later, we will see that when M = R, harmonie maps correspond to geodesics.

Classically we know they exist in closed manifolds. In the rclated problem of finding

harmonie representatives of de Rham cohomology classes of compact manifolds,

Hodge Theory asserts the existence of sueh representatives. It is interesting to note

that the same variation~l methods which we will develop could be brought to bear

on this problem as well, since a metric on the tangent bundle induces metrics on

the extcrior powers of the cotangent bundle.

So we see that a similar problem, that of finding harmonie representatives of

deRham cohomology classes, always has a solution for compact M. This is not the

case for our problem. The simple st ex ample (see [Eells-SampJ) of a homotopy class

of maps which does not admit a map of minimal energy is the cladf: homotopie to

id : S3 -+ S3. In this homotopy class of maps, Wf' c:tn construct maps of arbitrarily

( small encrgy, while we know a map with zero encrgy must be constant, and hence

9

not homotopie to id.

1.2 (Counter)example.

The process of suspension is a useful way of defining maps of Euclidean spheres,

which have a special structure. In this section, we show how the energy density, and

hence energy, of the suspended map may be calculated, in tenns of the previous map,

and exhibit maps S3 -+ S3 with arbitrarily small energy, which are homotopie to

the identity. Since no such maps can have zero energy, this homotopy class eonta.ins

no minimising harmonie map. ActualIy, homotopy classes of maps sn -+ sn can

be identified with their degrees. Our proof shows that for n ~ 3, only the nulI-

homotopy class contains an energy-minimising harmonie representative, i. e. the

constant map.

sn+! ca.n be thought of as the cylinder sn X [0,71"], with each end identified (or

'pinched'). In such a description, one has coordinate patches U x [0,71"] -+ sn+l

mapping (u,O) H (4J(u),O) E Sn X [0,71"], for each coordinate patch U ~ sn on sn.

The usual metric 9 = 9sn+1 on sn+! is given in these charts by

where hl) gives the metric on 4J(U). A map f : sn -+ sn can be suspended over a

map 'ljJ : [0,71"] -+ [0,71"], which fixes 0 and 71", by pinching the ends of the map f X lp.

Note that if f is of degree k, so is its suspension. CalI the suspension f.p.

10

(

We see

e(f", )(x, 9)

If we consider the map SI !:.... SI, z ~ zk, for k an integer, with energy density

e(k) = P/2 and suspend this map over id: [0,7r) --+ [O,7r), we get a degree k map

S2 -t S2 with energy density e(fk,2) = (k2 + 1 )/2. Suspending the result over

id: [O,7r] -t [0, 7r), we inductively obtain a degree k map A,n : sn --+ sn with energy

density

( ~ )_P+n-l e Jk,n - 2

The way to obtain maps of lower energy, will be to choose maps [0,7r] -t [O,7r]

which concentrate at 1r. Such a map is "p( 8) = 2 arctan[c( tan 8/2)k] with c > O.

The suspension A.c : sn+1 -t sn+! has the effect of pushing most of sn+! into a

small neighbourhood of one of the poles. To calculate the energy of fk,c, we need

~ sec2 "p( 8) "p' (8) = 2 2

d "p(8) 1 k-l 8 2 (J d8(tan-2-) = 2cktan 2 sec 2 1 1jJ l8 28

= - k tan - tan - - sec -2 2 2 2

=> "p'(O) =kcos1jJ/2 sin1jJ/2 = ksin1jJ cos 8/2 sin 8/2 sin (J

which allows us to compute

11

-,

l1r i [1 sin t/J 2 sin

2 t/J P + n -1] - -2(k--;-n) + . 2 B 2 dX(sin OlS" dO o (sin OlS" sm 17 sm

1 11r sin 2 t/J - -2 vol(sin osn)(2k2 + n -1) . 2 8 dO o SIn

- ~(2k2 + n - 1 )vol sn fo1r sin2 "p sinn-

2 0 dB

Lemma 1 J; sin2 'ljJ( O)dB ~ 0 as c ~ O.

Proof. Let 0 ::; € ::; 71'. On [0,71'/2] tangent is monotone increasing, so

choose a C > 0 such that C( tan 1r-;/2)k ::; ~. If € < 71', then

71' - €/2 tan 2 > 1.

For 0 ::; 9 ::; 7r - ~, and 0 < c ::; C,

So

o ::; sin2 arctan( c( tan ~)k) < sin2 aretan( C (tan 1r-;/2)k) < aretan(C(tan 'If'-;/2)k),

(sinee arg ~ 0) ::; €/271'

11r l1r-~/2 l'1f' € € sin2 t/J(O)d9 = ( + ) sin2 t/J(O)dO < 71' . -2 + -2 = €.

o 0 1r-~/2 71'

Since € was arbitrary, Ir: sin2 '1/;( O)dO -+ 0 as c -+ 0 as required.

Renee as c --+ 0, E(fk,c : sn -+ sn) -+ 0, for k an integer and n ~ 3. Therefore

inf{E(f) 1 f : sn -+ sn of degree k} = 0

for aU n ~ 3 for all k. Since deg(fk,c) = k, the homotopy class of fk,c contains no

zero-energy maps when k is not zero. It follows that when n ~ 3, only the trivial

12

( homotopy class of maps sn -+ sn has an energy-minimising, harmonie representa-

tivc.

1.3 Tension Field.

As in calculus of variations, we look for extremals of the energy functional as solu-

tions of Euler-Lagrange equations. These equations can he written as

rU) = 0

where rU) is a vector field along J, i.e. a section of the pull-hack bundle f*T(N)

over M, whieh has the property that

VvE(f) = - < r(f),v >,= - lM < r(f)(x),v(x) > dXM

where, for each p E M,

and v is a section of f* T( N), i. e. r(f) looks like the gradient of E on the space of

maps Coo (M. N) at the point (i. e. map) J, w here the sections of f* T( N) play the

role of the tangent space.

vVe can use this property to get r in local coordinates:

Lemnla 2 Let Jt : (M,g) --. (N, h) be a smooth Jamily oJ map8 for to < t < il'

Then

( where Nr and h are evaluated at f(p) and everything else at p.

13

Note: if v = ~(Jt) It=o, then eXPf(p)(tV(p)) is a first-order approximation of ft(p) at

t = O.

In local coordinates, the equations

are elliptic. In fact, if we take (Riemannian) normal coordinates at p and f(p), then

so that in these coordinates, each component of ris the Laplace operator applied to

that coordinate function. This ellipticity is used to prove many smoothness results

for solutions of r(J) = O.

Invariantly, thinking of dj as a section of T* M ® j*TN, we know how to take

its covariant derivative:

called the second fundamental form of f. The tension field, rU) is the trace of

this section, as one can verify by writing everything out in coordinates. A map f

is called totally geodesic if V( df) = 0, which certainly implies j is harmonie.

1.4 Geodesies.

In the case of curves I !... N,

14

( If we define the length of J, as usuaI,

LU) = fol 1 dt 1 dt,

we see L(J)2 < E(J), by the Schwarz inequality, where equality holds if and only

if 1 ~ 1 is constant (if and only if J is a constant speed curve). Hence if h is a

(1) (u) minimal geodesic, and h(O) = f(O), h(l) = J(1), E(h) = L(h)2 ~ L(J)2 ~ E(J)

and equality holds if and only if (i) J is of minimallength and (ii) f is a constant

speed curve, i.e. if and only if f is a minimal geodesic.

Henee, E : COO(I, N) ~ R takes on its minima on the minimal geodesics (see

[MilnorJ). Locally, the tension field is

(J)"( - d? j'Y N-n"( dfa dJP

'T - dt2 + l ail dt dt

for, = 1 ... n-the geodesic equations.

In faet, geodesics are precisely the extremals of the energy functional. Of course,

this says nothing about the existence of geodesics, let alone minimal geodesics. One

(though eertainly not the first) way to show the existence of such geodesics is with

classical Morse Theory. This follows since E is a continuous, positive functional on

the path space (of piecewise smooth paths), and we can show that only completely

smooth paths are minimal. Moreover, given €, sui tably small, the space of paths,

{f 1 E(J) < €}, can be approximated by a compact set; hence E must attain its

minimum (on a minimal gcodesic). Actually, Morse Theory tells us much more,

decomposing the space of loops into a CW complex (see [MilnorJ).

15

" In another section, we will see how classical Morse theory was generalised by

Palais and Smale and used by Uhlenbeck and later Sacks and Uhlenbeck, to gct

existence results for harmonic maps.

1.5 Surfaces.

Consider now the case where dimM = m = 2, i.e. when M is a closed Riemann

surface. As usuaI, we wouid like to know classes of maps which have minimal-en<,rgy

or harmonie representatives. We have already seen that with curvature restrictions

on N~ these representatives can be found in 'stable equilibria' of an energy-type

action. There are also several resl' Its which guarantee such representatives givcn

restrictions on the topological structure of N or of the representative, f : AI ~ N,

of the class of maps. In particular, Lemaire, and Schoen and Vau, and Sacks and

Uhlenbeck have shown that if 7l'2( N) = 0, then every homotopy class of maps has

a rninimal-energy representative (whieh is harmonie). If 7T'2(N) =1= 0, then thcre

even exists a generating set for 7l'2(N) consisting of conformaI branched immersions

whieh minimise energy in their homotopy classes (see [Sacks-U]).

We will relate the harmonie map problem to the Plateau problem ( as in [EeUs-Sam pl),

with a view to a strong regularity result of Saeks and Uhlenbeck:

Reeall that <p : M' ~ M is conformaI if there is a map fJ : M ~ R such that

<p*g = exp(2fJ)g'j i.e. in local coordinates,

-..........

16

, (

!

Actually, exp(29) could be any smooth positive function on M, but it will be con-

venient to assume this form.

If (x', y') are orthonormal coordinates at p (i.e. g'J = t5,j), then cf> is conformai

if and only if dcf>( :x) = ~ and ~ E T( M) are orthogonal, of the same length (and

nonzero). Allowing singular points, where ~ = 0= ~, we have the definition of

weak conformality. Now we see

e(J 0 cf> )(p)

- exp( 28) e(f)( cf>(p»

In particular, when M' = M, if cf> : M -+ M is a conformaI diffeomorphism, then

= exp(mfJ) dXM

80 E(fo cf» = JMe(f 0 1» dXM = JM exp(2fJ)e(f)(cf>(p» dXM but cf>: M -+ M is a

diffeomorphism, 80 E(focf» = fM e(f)(cf>(p» c/>*dXM = E(J). Hence E i5 a conformai

invariant. Note: this i8 valid for surfaces.

17

80 what is the relationship with the Plateau problem? What is the Plateau

problem? The Plateau problem is concerned with minimising the area functional

V( </» of an immersion of a manifold,</>: M -+ N,

or at least finding an extrema! map, which is called a minimal surface, and is

characterised by the equation T(</» = o. Note that the (intrinsic) metric on M

has no effect on area, which is determined entirely by the pulled-back metric, or,

in essence, the embedding. However, isothermal coordinates make thillgs more

transparent, and do not restrict our results in any way sinee Chern (in [Chern])

showed that we can always find a local isothermal coordinate system for M -that is,

a coordinate system (x, y) such that (</>*h)'J = À (~ ~) where <jJ*h'J is the metric

enduced by the embedding. In this coordinate system, V( </» = fM 1 À(x) 1 dX(M,q,·h).

Now put a metric 9 on M.

Lemma 3

V(</» ~ E(</»

and equality occurs if and only if </> is conformaI.

Proof.

V(<jJ) = lM 1 det(</>*h,J ) 11/2 dx, and

E(<jJ) = ! f (</>*h ll + <jJ*h22 ) dx. 2JM

18

( In orthonormal coordinates on (M,g) at p, chosen without loss of gen-

erality 50 that </J is orientation preserving,

det( </J* hl) )1/2 = (</J* hll ifJ* h22 - ifJ* h12 </J* h21 )1/2

< 1 </J*h ll 11/21 ifJ*h22 11/2

5 ~(I ifJ*hu 1 + 1 </J*h22 /)

= e( ifJ).

Integrating we obtain V( </J) 5 E( ifJ). Moreover, V( </J) = E( </J) if and only

if ifJ*h 12 = 0 = ifJ*h21 and </J*hu = ifJ*h22 at p, which is true if and only if

</J is conformaI, since p was arbitrary, as required.

As we noted, area is independent of the intrinsic metric on M. Energy, however,

is not. The minimum energy of a fixed map f under variation in the metric on M

is the area, which is realised by any metric conformaI to the metric induced by f.

Using the fact that E is a conformaI invariant, if j is a conformaI immersion, j

is harmonie if and only if j( M) is a minimally immersed surface. (Recall that an

immersed surface is minimal if and only if it is an extremal of V(f) with respect to

variations of f, and V(f) = E(f) for conformaI maps.) This gives a nice theorem

describing the surface, but first we need sorne c:efinitions and a result from [G-O-R]:

A map j : M -+ N, dimM = 2 is said to have a branch poznt at p of order k - 1

if there exists a k ~ 2 and local coordinates, Xl, x 2 on M and yI, ••. yn on N, such

19

that, with the convention that z = Xl + ix2,

yI 0 f(z) + i y2 0 f(z) = zk + O"(z)

y' 0 f(z) = e,(z) j = 3,'" n.

where 0" and e, are Cl functions satisfying O"(z), eJ(z) = 0(1 z Ik) as 1 z l-t 0;

î;'(z), ~(z) = 0(1 Z Ik-l) as 1 z 1--+ o.

Clearly, then, branch points are isolated. If in addition f is Cl in a neighbour-

hood of p, then f can be expressed as yI 0 f( z) + i y2 0 f( z) = zk where z --+ Z is

a Cl diffeomorphism, however, z will not in general be a chart for M unless !v! is

onlya Cl manifold.

A branehed immersion is a Cl map f : M --+ N which is regular (C k ) on

M - {Pt,'" pt}, where Pl,'" Pl are a finite number of branch points of positive'

degree.

The next result (also from [G-O-RD gives us a picture of what a harmonic map

can look like:

Theorem 4 If f is harmonie and weakly eonformal, then f zs a branehed zmmer-

sIon.

Combining this result with

Lemma 5 If f is harmonie, then <p is holomorphie, where

{ ôf 2 1 al 12 . af af) }d 2 <p = 1 ôx IT(N) - ay T(N) +2z( ax '8y T(N) z

20

(

(

is a quadratic, complez differential on M, ezpressed in isothermal coordinates on

M, z = Xl + ix2•

The vanishing of 4> is equivalent to the weak conformality of f. We see

Corollary 6 If f : S2 -+ N is harmonie and n > 3, then f is a GOO conformai

branched minimal immersion.

Proof. Sincf; fis harmonic, 4> is holomorphic, and hence zero, since the

Riemann sphere has no non zero quadratic diiferentials. Now apply the

previous lemmas and regularity result Cf harmonie implies C<YJ).

For surfaces of higher genus, we have

Theorem 7 If f is a critzcal map of E with respect to variation8 of f and of the

conformai structure on M, then f is a conformai branched minimal immersion.

21

Chapter 2

N onpositive Riemannian Curvature.

We now set up the framework for perhaps the first significant existence theorem for

ha.monic maps

Theorem 8 If Riem(N,h) ::; 0, and M and N are compact, then every homotopy

class of maps has a minimal, harmonie representative.

Lemma 9 If <f> : N -+ NI is a Riemannian immersion, then for f : M -4 N,

E(f) = E( <1> 0 1), and TU) = 1r( T( <f> 0 f)) where 1l' : <I>*T( Nd -4 T( N) is the bundle

projection of the pulled-back tangent spaee onto the tangent spaee of N (cano'll,ically

embedded in <f>*T(Nd).

Proof. We compute

e(<f>of) - !<9,(<f>of)*hN1 >

- ~ < 9, rhN > (since <f> is an immersion)

- eU)

22

Since 1> is an isometric immersion, we can identify T(N) with a sub-

space, 1>.T(N) of T(Nd, or we could identify J*T(N) --+ M with a

subbundle of (1) 0 J)·T( NI) --+ M.

In this description let 7r be the orthogonal projection onto T(N) from

rjJ*(T( Nd). We know that \1 N = 'Ir 0 V' NI (sinee 1> is isometric).

Pulling this structure back to (4) 0 1). T( NI), we make the following

calculation:

We know dJ maps T(M) into j*T(N), so df has a representation as si®v,

where s' E T*(M), Vj E j*T(N), i = 1· .. m. Since we are identifying

j*T(N) and (4) 0 J)*(f.T(N)), df = d(rjJ 0 f) in this representation.

Since T = tr(3, where (3(f) = \1(df) is the second fundamental form of

the immersion, we compute

(3(rjJ 0 f) = V TO(M)®(4>oJ)'T(Nl)(SI ® Vi)

= VTO(M)s' 0 Vi + s' 011" 0 VVi + s' 0 (V' - 7r 0 'V)v, (sinee V, E J*T(N) and 7rV, = Vi)

= VTO(M)® rT(N)df + SI /&) (V - 'Ir 0 V)Vj

hence (3(4> 0 J) = (3U) + terms perpendicular to j"'T(N) so (3(f) =

7r 0 f3( rjJ 0 1), which implies TU) = 1f 0 TC 4> 0 f), as required.

Corollary 10 f : M --+ N is harmonie if and only if T( rjJ 0 f) is perpendicular

ta j*T(N), or equivalently, ~f and only if T(rjJ 0 f) is perpendieular to 4>.T(N) in

23

T(Nt).

Eells and Sampson prove the following results (in [Eells-Samp]):

Theorem Il IJ Riem{N,h) ~ 0 and ft is a bounded solution to Vl- = r(lt) for 0 <

t < 00, then there exist 0 < t l < t 2 •• " sueh that {ftk} converge uniformly along

with their first order space derivatives to a harmonie map f, and f is homotopie to

10 and E(f) < E(fo).

Theorem 12 If Riem{N,h) ~ 0 and N is compact, then any Cl map f : Al ~ N

admits a unique solution ft to r(ft) = Vl-, fo = 1 for tE [0,00) such that f and its

first-order space derivatives are continuo us at t = O.

Hence when M and N are compact and Riem (N,h) ::; 0 every homotopy class of

maps has a harmonie representative. So if J. is any sequence of maps M _ N, which

tend to the minimum energy and, for each i, f: is the harmonie map dctermincd by

the above theorem, then lim E(f:) ~ lim E(f,)= mf{E(f) : f ~ fol. The sarne

estimates which give us the preceding theorem also give us that J: has a sl1bseqllcncc

converging uniformly along with its first derivatives.

To prove these results, Eells and Sampson use an embedding of N into R'I. An

embedding theorem of Nash makes this possible. This gives an isomctric cmbed­

ding, although Eells and Sampson show that it suffiees to consider any tubular

neighbourhood of Nin Rq with a suitable Riemannian-fibred structure. If wc stick

to the isometric embedding, we have shown that this embcdding does not effect

24

(

(

the encrgy of the map, and if ~ is the embedding, <p : N ~ Rq, then TU) = 0 if

and only if T(~ 0 f).l..T(N) ~ T(Rq). Analogously, (81at - T)ft = 0 if and only

if (ô 1 Bt - T)( ~ 0 ft ).l..~*T( N) <-+ T(Rq). One thus replaces the first elliptic system

wi th a parabolic system, such that the target manifold is now contained in one fiat

coordinate patch. Of course one needs to prove that a solution of the embedded

equations remains in N. Having proven this, EeUs and Sampson derive derivative

estimates depending on the curvature of N, as weIl as the compactness of M and

N. Since Rq is fiat rU) = 6(f), (the Laplace-Beltrami operator). Rence the main

equation looks just like the heat equation. And once they have obtained this critical

map, they apply standard PDE thechniques, in the guise of

Theorem 13 If f E C2(M, N) and rU) = 0, then f is smooth.

whose proof exploits the local form of TU) = O. A complete account of this ap­

proach, including aU the necessary PDE theory, and an extension to the case where

M has a boundary, can be found in [Hamilton].

25

Chapter 3

The case dim M 2.

Deformation by heat flow is eertainly a powerful tool for proving the existcncc of

harmonie maps. It is also the most straight-forward-almost intuitivc-mcthod wc

have so far. Unfortunately, it does not 'solve' the harmonie map problem (not that.

we can really expeet to). In connection with geodesics, we mentioncd 1\10rsc thCOl'y

as an approach to variational problems. In this theory, we shift the foeus from

the manifold to the space of maps. U nlike the last method, howevcl, this mcthod

requires the introduction of exotie spaces, a considerable inerease in sophistication,

which can be justifi.ed by considering what goes wrong in trying to gcncralise thc

various methods of showing the existence of harmonic funetions, to the case of maps

into manifolds with curvature.

Of the many methods for solving the first problem, Perron 's mcthod is thc must

elementary, and hence probably the most satisfying. In this methoù we eOllsidcr

subharmonic funetions (functions u on D such that 6.u ~ 0) satisfying the specifiee!

boundary conditions and take the pointwise supremum of subharmonic functions.

Subharmonic CreaI) functions on general manifolds ean of course be definecl in tenns

26

( of the generalised Laplace operator, but cannot be defined for maps between man­

ifolds, hence we cannot expect to generalise this method. Another method, which

readily generalises is the Functional Analysis, or Generalised function approach. In

this approach (to the solution of general PDE's), we look for solutions in terms of

distributions, or various subspaces thereof and try to construct a regularity theory,

which tells us when a solution in the generalised sense (of distributions) actually lies

in a smaller space (of smooth functions, for example). For this, we will introduce

Sobolev spaces, although some approaches require other spaces also used in PDE

theory such as HoIder or Schauder spaces.

The drawback of resorting to Sobolev spaces is that we have to sacrifice geometric

intuition when working with maps which are not continuous. So things which make

sense classically, cannot be translated into this new language. Sorne geometrical

content is preserved, howeverj for example, we will define a ?rI-action induced by a

map f E L~(M, N), which a priori is defined only for continuous maps. This is a

surprising result, which points to the existence of harmonie maps representing each

?rractioIl.

3.1 Sobolev Spaces.

For 1 $ p $ 00 and k > 0, we define LJ'(M,Rq) to be the space of measurable

functions M -4 R q which are bounded in the norm

27

and LHM, Rq) to be the measurable functions bounded in the norm

where Da is the distribution theoretic derivative, a is a multi-index and

in sorne coordinate system. (Changing coordinates gives equivalent norms, hence

den.nes the same Banach space topology on Lt(M, Rq).) Of course, strictly speaking,

the resultant norms are really only semi-norms, but as usual we quotient by the'

functions of norm zero and obtain a true Banach space. When we speak of an Lr function, then, we are really speaking of an equivalence class of functions. In the

case k = l, note that Il . 1 h,v is equivalent to Il f Ilv +(fM 1 di IV dx M )1/v, where 1 df 1

is tpe norm on the differential induced by the norms on Rq and T*(M). Similar

equivalent norms can be formulated for higher dimensions.

For N C Rq, L1(M, N) will be the space of maps in L1(M, Rq) such that almost

all x E M have images in N, given the subspace topology. This is not always

a closed set. In particular, if p = 2, L~ is a Hilbert space, with inner product

Sobolev spaces exhibit a certain regularity, which may seem surprising ta some-

one not familiar with them. We already know ways of approximating measurahle

functions by continuous ones (Lusin's Theorem, for example). The Sobolev Embcd-

ding Theorem

28

(" Theorem 14 If f E L~(n, R), n c Rm, and 0 ~ 1 < k - m/p for SOThe l, k E

NU {O} then f can be represented by some 1 E C/(O, R).

tells us that Sobolev norms, defined entirely in terms of measure and adjoint-

dcrivatives, can be stronger than the uniform Ck norms. The inequality

1 < k-m/p

puts limits on what we can expect from the Sobolev-space methods. In the case

m = dim M = 1 (the case of geodesics) the Sobolev inequality tells us that elements

of L~(M, N)- the natural space to consider when dealing with E, defined in terms

of first-order derivatives- are continuous. Unfortunately, when dim M = 3, we

do not even know whether these functions are continuous. This is the principal

obstruction to extending the methods to higher dimensions. For completeness, we

state a more general form of the Embedding theorem:

Theorem 15

3.2 Stretching the Sobolev Inequality.

As promised in the introduction, we can expect results when restricting the homo-

topy type of a map or manifold. Continuous maps between manifolds induce cor-

responding maps between homotopy groups. The question is, can one extend this

( notions to Sobolev spaces of maps? Schoen and Yau (see [Sch-VauD showed that we

29

can, in fact, induce a well defined map 'Trl(f) : 'Trl(M)_'Trl(N) when 1 E Li(AI, N)

and M is a surface. 1 They give the following proof, which we will apply in tlw

following section to get an existence theorem for harmonie maps.

We know that 1 E L~ is not al ways continuous, but wc know that away from a

set of arbitrarily small measure, it is continuous. Given (J E 'TrI (AI), a loop in .A1,

the composition f 0 t7 is not necessarily in 'TrI (N), as it is not necessarily cont.inuous!

We include the proof because it shows how concepts which seem inherclltJy t.o

involve the continuity of a map can, with ingenuity, be extended to maps in Sobolev

spaces which are 'close to being continuous', in the sense that the Sobolev incql1alit.y

is almost satisfied. The proof depends essentially on the lemma

LeInma 16 Let D be the unit dise in R2 and f : D -+ N be an Li map into a

compact manifold. Let Cr (0 < r < 1) be the circle in D of radius r about the

origin. Then I( Cr) is contractible for almost every r E (0,1].

The pro of will depend on results from minimal surface theory, as weIl as regulal'i ty

results. Both these types of results figure prominently in the various strategies for

showing harmonie maps of surfaces exist.

Proof. Consider N to be isometricaIly embedded in Rq. Since it is

compact, the tubular neighbourhood

T = {x E Rq : di.st(x,N) < h}

IThis result generalises to higher dimensions, and correspondingly sm aller Sobolev spaces. Sec [Burstall] and [White].

30

« of Nin Rq is diffeornorphic to the normal bundle of Nin Rq, for SOIlle

b > O. Renee, there is a well-defined projection c:>f T onto N which

is a homotopy equivalence. Now, f E L~(D,N) and the compactness

of N imply EU) < 00. SO ID 1 df 12 dx < 00. But ID 1 df 12 dx

= fiUcr 1 df 12 dxcJ r dr with the induced measure on Cr. Since

1 E Li{D, Rq), Je .. 1 df 12 dXCr < 00 for almost every r E (0,1]. Hence,

since 0 ~ 0 < 1- 2;1. We know, then, that for almost all r E (0, IJ, 1 ICr

is continuous and we will show that for such r, f( Cr) is contractible in

N.

Before we continue, we state a technica1 result in minimal surface theory

from [Rass-Scott, 2.3], which we will need.

There exists sorne Q > 0 and fi > 0 such that if A is a least-area disk in

N, Xo E A andr < a, then 8AnBr (xo) = 0 implies area(AnBr{xo» >

fi r 2• If a and fi are such numbers, we may assume without 10ss of

generality, that a > 8, the radius of the tubular neighbourhood.

Now let ro be such that f is continuous on Cro, and small enough that

! r 1 dl 12 dx < {3(~)2. 2 J{xED:lxISro} 2

Then area(f( Dro» < f3(!})2. (Remember that area Îs bounded by en-

( ergy. See Lemma 3.) We proceed in two steps:

31

Step 1.

We know that ero spans a unique solution, h : Dro --+ Rq of the equation

b.h = 0, h = f on Gro and that h is smooth on Dro and continuous on

Dro compact, see for instance [Hass-Scott, 8.3,8.11,8.30]. Now if we had

0, sinee h(Cro) c N. But then area(h(Dro)) ~ area(h(Dro) n B/i/2(XO))

> (3(8/2)2, which would contradict our earlier estimate. Hence h(Dro) C

T, so h( Cro ) is contractible. In fact, this holds for almost any r :::; ro.

Step II.

Now, if we choose To < Tl < ... < Tl = 1 such that

and f is continuous on Cr. for all i, the trick we used to show J was

continuous on almost every circle also shows that J is continuous on

almost every ray, and given a ray rand r, < r < r'+l

we can construct a disc D' spanning Gr. and Cr.+!, by which we mcan a

spanning annulus, which becomes a disc whcll we removc a radius. By

step 1 D' is contractible in N,50 J( Cr.) is freely homotopie to f( Cr) in

N.

Our result follows by induction on i.

32

(

Figure 3.1: Contour of Integration

Corollary 17 If f E L~(Drl - Dro,N), Tl > TO, then fOT almost all T,S E (rO,r1),

f(Cr ) is homotopie (freely) to f(C$) in N.

N ow, if M is a Riemann surface and Il,·· ·"Y2g : SI --+ M generate ?rI (kf),

construct tubular neighbourhoods, 11 of li in M with coordinates given by 'ljJi :

SI x [-1, 1] ~ 1: where"pt IS1X{0}= "Yi. Furthermore, assume the "pt agree on a

neighbourhood, U, of {O} x [-1,1]. Now f E LUM, Rq) implies f 0 .,pt E Li(Sl X

[-1,1], Rq). Using a diffeomorphism SI X [-1, 1] ~ Dl - Dl/2' the corollary gives

us a free homotopy class of loops in N to associate to each generator of ?rI (M).

We can, however, choose an So E [-1,1] such thatf o.,p, is continuous on SI X {sol

for all i, and a Bo such that f o.,pt is continuous on {Bo} x [-1,1] for aIl i, and

{Bo} x [-1, 1] C U. Without 10ss of generality, assume we can take 80 = O,otherwise

take "Y. = .,pi ISIX{!O}. It remains to verify that the map

(

33

defined on generators by

is well-defined.

To see that this is well-defined, consider a relation "I~II/f; ... 'Y~I = e in 7r} (111, *).

The image of a homotopy (i. e. contraction of the loop 'Y~11/~22 ••• 1':/) is a di sc in M

whose boundary is mapped eontinuously into the loop 1 = j( "1:11 )j( 1':22

) ••• j( I~,') in

N. Since f E LHM,Rq), f is L~ on the dise, and by the lemma, 1 is contractible.

Renee j defines a map 7r1(f) : 7r}(M) -+ 7rl(N), as required.

3.3 The Application.

Raving thus defined a 7r1-action for L~ maps, it is easy to show that harmonie

representatives exist for maps of surfaces inducing the same 1l'}-action.

Let jo E COO( M, N) be a fixed map. Let

Fa = {j E L~(M, N): '/rt(j) = 1l't(O'-l 0 fa 00'),0' E COO([O, l],N),

= {f E L~(M, N) : for sorne path 0' connecting f( *) to foC *), the loop

(7-1 0 Jo 0 0' induces the same map on the fundamental groups}

= {f E L~(M, N) : f and fa induce the same map on the free

homotopy groups}.

Note that Fo is not a homotopy class in general. Let m = inf{E(f) : f E Fo}.

34

-

( Theorem 18 There exüt~ an 1 E Fo &uch that E(f) = m.

Proof. Recall that LHM,N) was defined to be a subspaceof LUM,Rq),

making use of sorne fixed isometric embedding, N C R9. For conver-

gence questions, we will always work in the larger space.

Let J, be a sequence in Fo such that EUd ~ m, as i -+ 00. Since

{E(J,)} is bounded, and N is compact, {fI} forms a bounded set in

Li(M, Rq), hence there exists an 1 E Li(M, R9) and a subsequence

{fl} c {fI} which converges weakly to fin Li(M,Rq), and strongly in

P(M, Rq). Hence we can assume IJ ~ f pointwise almost everywhere,

so 1 E Li(M, N) since N is dosed. We daim f E Fo, and EU) = m.

We know there exists a k > 0 such that E(f,) ~ k for an j. Lower

semi-continuity of E gives us

EU) < lim E(f,) = m. 3 ..... 00

See [Morrey2, Thms 1.8.1,1.8.2J.

Recalling the notation of the previous section (page 33),

J ( 1 d(J, 01/J,) /2 de dt < /(1 lSl X[-l,l]

for aIl i,j. By a now-familiar argument, Il is continuous on Il,8 (=

.,p, ISIX{,,}) for aIl i and j, for almost every s E [-1,1]. For one such s,

there exists an AI" > 0, such that

35

for infinitely many j, for aU i. These infinitely many j form a further

subsequence, which is equicontinuous on /1,6 for aH i. Since N is com-

pact, we can apply Ascoli's theorem f.o find aj such that f J is arbitrarily

close, in the uniform norm, to f on "'Ii,s, for i = 1,2,'" 2g. From which

it follows, that f induces the same free action on ?rI (M) as this f J , hence

f E Fo, as required.

Of course, a priori, we know nothing about the smoothness of this map, which

again follows from one of many regularity results, such as

Theorem 19 If f E LUM,N) is an energy-minimising map with f(x) E No for

almost every x E M, for some compact set No C N, then dîmeS n interior(M)) ~

m - 3, where S is the set of singular points on M (see [Sacks- UJ).

Here dim refers to the Hausdorff dimension of a set, given by

dimX = sup{p: mp(X) > D}

mp(X) = sup{m~(X) 1 € > D}

00

m~(X) = inf{2::(dîamA,)P 1 X = U~lA" diama, < e} 1=1

In particular, if N is a 3-manifold, then f is smooth except on a discrete set.

36

Chapter 4

The case dim M o.

The third approach to the existence theory we will consider proceeds via Morse The-

ory, and the perturbation method of Uhlenbeck. Questions about maps between

manifolds can often be phrased as questions about spaces of maps between mani-

folds, usually Sobolev spaces L~(M, N). Morse theor~- can be applied to such prob-

lems when the space in question is a closed manifold. Unfortunately, the Sobolev

inequality puts a restriction on when L2(M, N) will be a manifold on which E is a

Morse function. In fact, when dim M = m 2:: 2 it is not.

Section 4.1 briefly describes Morse Theory and the generalisations which allow

us to consider Banach Manifolds, and introduces the idea of using a perturbed

encrgy functional. In section 4.2, we con si der the perturb ~d energy

We caU this a perturbation because as a: ~ 1, EaU) ~ EU) + 1. The main

work in this approach, is to show that critical values of Ea, a > 1 approximate

critical values of E (for small a). The bulk of this section is devoted to showing

37

- Laa 2 22Mil8Q) Qi i i 2 W li !IL a j

that Ea-minimising maps fa, which converge weakly in LUR2, N) converge locally

uniformly in the Cl norm if 1 dfa 1 is locally uniformly bounded, and that otherwise,

the convergence process bubbles off a sphere, i. e. that convergence is blocked by an

incompressible, embedded 2-sphere, which is just a generator of 7r2(N)). Finally,

we use this to show that when 7r2(N) = 0, minimising harmonie maps do exist.

4.1 Morse Theory.

Morse Theory, in its simplest form, tries to reconsturct the topology of a compact

manifold embedded in a Euclidean space by considering the critical points of its

height function in general position, i.e. with isolated, nondegenerate (full-rank Hes­

sian) critical points. Palais and Smale generalised it to consider Hilbert manifolds,

notably, spaces of functions, with a view to applications to variational problems

such as ours. The point of the theory is contained in their main theorem:

Theorem 20 Let M be a complete Riemannian manifold (i.e. a Hilbert manzfold

with admissible metric) of class c k+2 (k ?: 1) and f E Ck+2(M, R). If f has only

non-degererate critical points, and if

(C) Any subset S of M on which J is bounded and 1 df 1 is not bounded away from

zero, has a critical point in its closure.

Then

(a) the critical values of f are isolated and there are only finitely many critical

points at each level, i.e. in f-l({ c} );

38

.lia 2

f ,

(b) if [a, b) eontains no critieal values, Ma = {x E M 1 f( x) ::; a} is a deforma-

tion retract of Mb,'

( c) if a < c < band c is the only critical value in [a, b] and pl, P2 ... Pr are

the eritieal points eorresponding to e, with finite index kl respectively, then Mb is

homotopie to Ma with kl-eells attaehed, i = 1,2,' .. , r.

Assume for the moment that f : M -+ N is a Coo map and that the space of

such maps has the structure of a Hilbert manifold, on which E is continuous and

satisfies (C), then the subspace of paths homotopie to f -the component of f-has

a minimal energy, harmonie representative. Unfortunately, these conditions are not

even satisfied when dimM ~ 2 since Lf(M, N) functions are no longer required to

be continuous, making nonsense of the notion of homotopy classes.

This, however, turns out to be an unnecessary restriction, since we can get by

with Ljusternik-Schnirelman theory, a weaker form of Morse theory which does not

require a Hilbert space structure on the tangent space (i. e. it does not require as

strong a nondegeneracy condition on critical points.) This allows us ta work on

other Li: spaces, and extends the applicability of these techniques.

For a complete account of Ljusternik-Schnirelman theory see [Palais2j. Briefly,

wc deal with Finsler manifolds, which are Cl, Banach manifolds with a map Il . Il :

T(M) -+ R such that II· Il : TAM) -+ R is an admissible norm. (An admissible

norm is one which cornes from the Banach space norms on the trivialisations of the

vector bundle T(M).)

39

....... __ ._-------_. ---_._-----------------------------.

Theorem 21 If M is a complete C2, Finsler manifold and f : M ---. R is C2- (i.e.

its first partials are Lipschitz) satisfying (C) and bounded below, then f has at least

cat( M) critical points.

where cat(M) = inf{n 1 M can be covered by n closed, contractible dises}. Hence

cat(M) > 0 for M =f. 0.

Of course, in our case, we are primarily interested in the existence of critical

points (and in particular minima) for f, not in the structure of the Banach manifold

COO(M, N). So this is almost ellough information already. However, a lot of analysis

is required to show condition (C) is satisfied for a given functional. (See [Palais3],

for example.) For E, condition (C) is dept:;ndent on m = dimM. For m = 1 (C) is

satisfied.

As a first test though, we would like ta know: Does this theory apply ta the case

already treated by Eells and Sampson? Ta which the answer (given by Uhlenbeck

in [Uhll]) is 'almost'.

In this case, however, it is not enough to consider the functional EU), as it

is not continuous on L~(M, N) for m > 1, because the Sobolev inequality is not

satisfied for m > 1, and do es not satisfy the condition (C) on smaller L~(M, N)

spaces. If we perturb E, however, by considering instead E(f) + € GU) where

G(f) = fM 1 df 12m dXM then, on L~m(M,N), E + €G satisfies (C) for all € > 0 and

the critical set S = {f E L~m(M, N) : d(E + €G)(f) = 0 and a < EU) + f GU) :5 b,

for an €, such that 0 < f < 8} of the family of functionals {E + fG 1 0 < € < 8} has

40

(

(

compact closure for sorne é > 0, where a < b not critical values. Moreover every

critical rnap, J, of E, satisfying a < E(f) < b, is contained in sorne neighbourhood

U of J in L~m( M, N), which has the property that the critical set of E + f..G in U (i. e.

the set Sn U ) is a 'curve' Jl (e E [0, é», Jo = J whose elernents are nondegererate

with the same index.

In the next section, we will consider a different perturbation, sirnilar to this one,

but which is neither a generalisation nor a special case of this one.

4.2 Perturbation.

In this section we will assume dirn M = m == 2. Our goal will be to outline a proof

of the

Theorem 22 If M and N are compact, M has a fixed conformaI structure, and

7r2(N) = 0, then for aU r E 7l"0 CO(M, N), there exists a minimising harmonie map.

Note that this theorem is also a consequence of theorem 18.

To do this, we define

EaU) = fM(l+ 1 df 12tdxM

for a 2:: 1. For compact M, Ck(M, N) C L~(M, N) c L'k(M, N) for 1 :5 a < p.

Ea will be evaluated on Lia functions, since Lia(M, N) is a C 2 separable Banach

manifold for a > 1 (see [Palais3]) on which

Theorem 23 For a > 1, EOt is C2 on L~a(M,N) and satisfies the Palais-Smale

condition (C) given a complete Finsler metric on Lia(M, N), when N is compact.

41

To show that L~Ot maps which are critical for EOt (a > 1) are smooth Sacks and Uh­

lenbeck determine the Euler-Lagrange equations for EOt and argue that smoothness

follows, as in the a = 1 case, for a > 1.

In the following, D(R) will be the disk in R2 of radius R about the origin,

D = D(l),and 1 • IO,k,p with denote the Sobolev norm on L~(n). Note that 1 f 10,1,2

and 1 di 10,0,2 both bound (a constant multiple of) the energy of f : n -+ N. The

basic analysis behind their work cornes in the form of an estimate

Theorem 24 For any .P E (1,00), there exi8t8 an € > a and an ao > 1 sueh that

for any 8maller di8k D' c D, any 8mooth eritical map,j : D -+ N, of EOt, with

EU) < € and any 1 ~ a < ao, there exist.9 a constant C(p, D'), 8uch that

1 df IV',l,p< C(p, D') 1 df IV,O,2 .

One ean use this estimate to get a bound on 1 df( x) 1 . 1 x 1 in terms of 1 df ID,O,2

for x E D, which leads to the important result

Theorem 25 If f : D \ {a} -+ N is harmonie with finite energy, then f extend8 to

a .9mooth harmonie map f : D -+ N.

This result can be seen as an analogue of the standard result of eomplex analysis,

which says that a bounded holomorphie function on D - {a} extends to a holomor­

phie function on D.

Now how do es this sort of result combine with the faet 7r2(N) o to give

existence theorems for harmonic maps?

42

r

Lemma 26 If (fa) is a sequence of critical maps for Ea, with EaUa) < B, for

some fixed B > 0, then there exists a subsequence (fJ) C (a) and an f E LUM,Rq)

such that fp -+ f weakly in L~(M, Rq) and limE(if3) ~ E(f).

Proof. Any closed ball in a Hilbert space is weakly compact, so find

a subsequence which converges weakly. The second statement is just

the Iower-semicontinuity of E. (For a proof, see again [Morrey2, Thms

1.8.1,1.8.2].)

Theorem 27 There exists an € such that if fa : D( R) -+ N is a sequence of weakly

convergent (in L~(D(R), Rq)) critical map8 for Ea, such that E(fa) < € then fa -+ f

in C1(D(R/2), N) and f : D(R/2) -+ N is a smooth harmonie map.

Now

Proof. Since E is a conformaI invariant, assume D(R) = D without loss

of generality. Choose € as in the main estimate (Theorem 24), with p = 4

and D' = D(1/2). Then 1 dfa ID(1/2),1,4~ C( 4, D(1/2))€ uniformly as

ct -+ 1. The Sobolev compact embeddingL~(D(1/2), RQ)CC1(D(1/2), RQ),

takes the bounded set {fa} into a set with compact closurein C1(D(l /2), Rq),

and so fa --+ f in C 1(D(1/2), Rq), sinee fa -+ f pointwise almost ev­

erywhere, a priori. And Sacks and Uhlenbeck obtain that f is srnooth

harmonie by examining the Euler-Lagrange equations for Ea and E and

using the fact that they are 'close'.

43

Lemma 28 Let U be an open subset of M, and fOi : U --. N C Rq be a seq1Lence

of critical maps of EOi for G -. 1, fOi -. f weakly in L~(U,Rq), and EOi(fa) < B for

some B > O. Let UJ = {x EU: D(x, 2-J+1) eU}. Then there e:wts a subseq1Lence

(j3) c (ct) and a jinite number of points {Xl ,J' ... XI,]} where 1 depends on Band N

Proof. Use the sarne € found above. Cover U] by disks D(xj,2-J ) CU,

such that eaeh point x E U is eovered at rnost k tirnes, for sorne fixed

k. Then

and for each a, at rnost Bk/e of the disks are sueh that

The previous theorern irnplies {fa} converges in C 1(D(x" 2-3), N) un-

less (*) holds for infinitely rnany a, which would contradict the weak

convergence of {fa}. Hence for aIl but sorne 1 < kB/€ + 1 dises, fa

converges in Cl.

The idea is that restricted energy rnaps on dises behave nicely. So maps of bounded

energy behave ni cely except in arbitrarily srnall neighbourhoods of bad points, where

the energy is not sufficiently restricted. Taking smaIler and sm aller neighbourhoods

leads to

44

(

(

Theorem 29 Let U be an open subset of M, and fcx : U -. N C Rq be a sequence

of critical maps of Ea for a -+ 1 and fa -. f weakly in L~(U, Rq), EO/(fO/) < B

for some B > O. Then there exisis a subsequence (f3) C (0:) and a finite number of

points {Xl,'" xL} such that f{3 -+ f in Cl(U \ {x!,'" XI}, N) and f : U -. N is a

smooth harmontc map.

Proof. 1 terate the last lemma to get a descending sequence of subse­

quences and take the diagonal subsequence in the usuaI way. CalI it

({3). We claim f{3 -. f in Cl(U \ {xt,· .. xd,N), for sorne finite set

{Xb"·X/}CU.

We know f{3 -+ f off sorne singular set. Assume, on the contrary, that

this set contains more than Blé points. Choose lBléJ + 1 such points.

If the minimum distance between any two of these points, and the min­

imum distance between any one of these points and a point outside U

is > 2-)+1, then any covering of U) by disks (as in the Iast lemma)

wouId contain l kB 1 f J + 1 disjoint bad (*) disks. This contradicts the

hypothesis EUcx) < EcxUcx) < B, which proves our claim.

By Lemma 26, E(f) ~ lim EUo) < B, so theorems 25 and 27 imply f

is harmonie.

Of course, the exclusion of the points Xl, ••• XI is necessary, and the convergence

cannot always be extended to these points.

45

We would like to know what sort of bad behavior a sequence can have at bad

points. We can imagine that something similar to the example in section 1.2 could

happen, if there were one bad point. In this case the whole sphere is pushed onto

one pole, but the image of each element in the sequence is the whole sphcre. The

limit is a point, with an adjacent 'ghost sphere'.

Let fa be a sequence as in the last theorem, and consider the following situation,

which answers the important question: When is a bad point really bad?

Theorem 30 If, in addition to the hypotheses of the last theorem, the1'e exzsts a

f> > 0 sueh that max{1 dfa(x) 1: x E D(xt,f>)} :::; B < 00. Then fa -+ f in

Then theorem 27 gives that fa -+ f in C1(D(XI, R/2), N).

So to be a really bad point, the derivatives of the fa must 'blow up' in a neighbour-

hood of Xl. However,

Theorem 31 If fa is a sequence of critical maps of Ea for Q -+ 1, fa f+ f in

harmonie map j : 8 2 -+ N which is not constant, sueh that

ICs2) c n (n U f(3(D(xI, 2-rn )),

rn-+co a-1 /3~Ot

46

( Proof. Let j > 0 be !arge enough that {Xl,' •• XI} n D(xt, 2-') = {Xl}

and so that exPX) : UeR 2 -+ D( Xl, 2-j) is a smooth bijection, for

sorne open set UeR 2• This bijection allows us to make use of the

vector-space structure of R 2 inside D( Xl, 2-').

Consider the maps fat ID(Xl,2-J)' 1 dfat(x) 1 takes on a maximum, be., at

sorne point, X'n and by the last theorem we know X Ot -+ Xl and ba -+ 00

when this makes sense, i. e. for smali Q' since xa -+ Xl, and ba -+ 00 as

a -+ 1. Then la is still a critical map of Ea, since we are essentially

composing with a conformaI map, but now 1 dlOl(x) 15 1 on D(O, 2- j bOt ).

In particular 1 dla(O) 1= 1. Since bOt -+ 00 as Q' -+ 1, conformaI dilation

X H b;;l X brings the metric closer to the Euclidean metric. Now fix an

R> O. For Q' sufficiently small, D(O,R) C D(O,2-Jba ), and]OI ID(Q,R)

satisfys the conditions of the Iast theorem, hence there exists a sub-

sequence, ({3) C (a), which converges in CI(D(O,R),N), (everywhere,

since 1 djOt(x) 1 is bounded uniformly). Since 1 dJOt(O) 1= 1, 1 diCO) 1= 1

also, hence ! is not the constant map. Since R was arbitrary, we can

assume! is defined on R2 = limR-+oo D(O,R).

Now

(

47

Setting m = -log bp, the LHS ~ E(Ï) + E(f) as {3 ~ 1. Hence

E(j) + EU) $lirni3-+1E(f{J).

Finally, R 2 = S2\ {oo} conforrnally, and E(j) < 00, so, by the extension

lernrna (theorem 25), j extends to a rnap j : S2 ---. N (with the sarne

energy), as required.

So if sorne sequence of critical rnaps of E~, f~ -+ f weakly in LUM, Rq), then

either fp ~ f in C1(M,N) for sorne subsequenee ({3) C (/1'),01 there is a non-

constant harmonie map j : S2 ~ N with Ï( S2) c na-+l Up~~ f{3( M). This latter

phenomenon, is sometimes called 'bubbling off a sphere', because, we imagine min-

imal E~ surfaces looking something like

0(::.\.\

Obviously, if 7r2(N) = 0, every sphere in N is contractible, so no nonconstant,

minimising harmonie embedded spheres exist. So bubbling off cannot occur. This

suggests

48

Theorem 32 If N is compact and 7r2(N) = 0, then every component of [M, N] has

a minzmi8ing harmonie representative.

Of course, the proof is not quite so simple, but the basic idea is already there.

Sketch of Proof. Fix a homotopy class r of maps. Since, for 0: > 1, Ea

satisfies the Palais-Smale condition (C) on Lia(M, N), and has srnooth

critical maps and CO(M, N) and Lia(M, N) have the sarne hornotopy

type (see [Sacks-U] and [Palais3]), we can choose a sequence fa of min-

imising maps for Eo in r, for sorne sequenœ a -t 1. Take any differen-

tiable rnap 1 E r (we have a whole sequence), and let

B = max{1 d/(x) 1: xE M}

(which exists sin ce M is compact), then clearly

vVe know there is a subsequence of f3's and a harmonie map f such that

We cau modify ff3 on a neighbourhood D(Xl' R), to form a new sequence

-1 - eXPJ( )

so that ff3 -t fin D(xt,R). Choose r > a such that D(J(xI),r) -t"'1

TJ(x])N is a injective (and hence a horneomorphism onto its image),

( which induces a local linear structure on DU( Xl)' r) (which of course

49

î J

J

j l

... -. is not unique). Fix R > 0 such that D(xt, R) n {Xl,'" x,} - {xd,

f(D(Xl,R» c D(f(xI),r/2) and

R ~ 1 < V 2; Il f 111.00

If TI is a Coo funetion R --. R, wi th TI ( x) = 1, x ;:::: 1, and." ( x) = 0, x < ~,

let

Then, if 1 X-Xl I<~, J(3(x) = f(x), but if 1 X-Xl 1= R, J(3(x) = ff3(x).

sinee 1 . 1: M ~ R is smooth on a smooth, Riemannian manifold,

j{3 is smooth. Moreover, for X i= XI, f{3 --. f in Cl(U, N) for some

neighbourhood U of x, with Xl fi. U. So j(3 -+ f in CI(U, N). Hence

j{3 -+ f in CI(D( Xl, R), N). Since 7r2(N) = 0, ff3 and jf3 are homotopie

(since they agree on {x :1 X - xII> R}, making f{3 ID(x},R) ujf3 ID(xl,R) ét,

sphere in N).

Now, notice that f{3 and j{3 agree on {x :1 X-Xl 1= R}. Gluillg f{J and j{J

on the boundary of D(Xl' R), we obtain a sphere embeddcd in N. Since

7r2(N) = 0, this sphere bounds a 3-ball. Hence, there is a homotopy of

f{3 ID(xltR) and j{3 ID(xl,R) fixing the boundary of D( Xl, R). Since fi3 was

chosen to minimise E{3 in its homotpy class, a.nd f{3 and jf3 agrcc outsidc

50

(

(

Whieh says that the derivatives of f/3 do not blow up sufficiently for a

sphere to bubble off (Theorem 30), hence, by induction on 1, 1/3 -+ 1 in

C1(M, N). 80 EU) = limp-+l Ep(fp), and f is a minimising harmonie

map.

51

Chapter 5

Conclusion.

So far we have defined the harmonie map problem, given two manifolds. Wc hr,"c

seen that this problem is-in a way-an extension of the geodesie and minimal sur­

face problems in differential geometry, and the links between minimal surfaces and

harmonie maps of surfaces are still alive. In faet, since Energy is a conformaI in­

variant, we found it convenient to eonsider the energy of a map as a function of the

conformaI structure on M. Sueh investigation led us to a strong regulari ty theoreIll,

of Sacks and Uhlenbeck, for maps of surfaces.

Since the existence of harmonie rnaps is a generalisation and an amalgamation

of several classical problems in mathematics, one might hope to apply the saIlle

classical methods in the more general setting. This, however, is not the ease, aH

dimension al problems destroy any such hope in several differcnt ways. VUHt amounts

of work have gone into 'updating' classical methods, such as Morse TheOl'Y, which

Palais remarked gave a very elegant proof of the existence of minimal ge()de~in;,

The souped-up Morse theory came complete with Hilbert and Banach spaces, aIlCi

sprinkled with functional analysis, but the conditions neecled for such a theory (for

52

(

(

examplc the Palais-Smale condition for E) are too restrictive, and we were left with

only scattercd results, and results only in low dimensions.

Unlike the situation in one dimension, (looking for (minimal) geodesics in the

spacc of paths) the various necessary conditions so far uncovered for the existence

of harmonie maps are quite complex and more restrictive. No one method or set of

necessary conditions can daim much universality. Instead, we have a basketfull of

results, of which We chose to describe three in sorne detai1: the first because it was

among the first ta deal with a large class of manifolds, and, historically, generated

a lot of interest in the field; the next because it illustrates the growing usefulness

of PDE theory (especially Sobolev spaces) in harmonie map theory, as well as it5

shcer cleverness and probable future significance; and the final because it illustrated

a significant enhancement of Morse-theoretic methods, which will certainly find

future application.

Although the methods vary widely, they illustrate several patterns

• a not-surprising reliance on analysis, in the form of estimates on elliptic (and

parabolic) systems of PDEs, which lie at the heart of aIl regularity results,

of the convergence of Eells and Sampson's heat distribution, and Sacks and

Uhlenbeck's perturbed minima;

• basic variational methods, eEpecially the existence of convergent subsequences

in coarser norrns (LV) given boundedness in fl.ner (Ln norms, and other uses

of Sobolev embeddings, and lower semi-continuity of E which show these

53

sequences converge to minima; and, finally,

• 'naive' variational methods. Given a Morse function on a manifold, Al, clas-

Gieal Morse theory uses deformations of level submanifolds by the gradient

field 'V f to decompose M. Ljusternik-Schnirelman theory first constructs a

pseudo-gradient field, and then does the same thing. EeUs and Sampson use

the same deformation to obtain minimal harmonie maps, as Milgram and

Rosenblum did in another case.

This last tool seems very simple, yet is behind sever al of our mcthods of find-

ing harmonie maps, and holds promise that we can further decompose the spa ces

L~(M, N), and Ckta(M, N), as classical Morse theory deeomposed n(M, p, q), the

spaee of piecewise-smooth curves in M, connecting p and q.

54

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58