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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
(
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
-
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.
(
7
-
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
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