Table Of ContentLecture Notes in
Mathematics
Edited by A Dold and B. Eckmann
471
Richard S. Hamilton
Harmonic Maps of
Manifolds with Boundary
Springer-Verlag .
Berl in . Heidelberg . New York 1975
Author
Prof. Richard S. Hamilton
Department of Mathematics
Cornell University
White Hall
Ithaca, N. Y. 14853
USA
Library of Congress Cataloging in Publication Data
Hamilton, Richard S 19~'-
Harmonic maps of manifolds with boundary.
(Lecture notes in mathematics; 471)
Bibliography: p.
Includes irdex.
1. Global analysis (Mathematics) 2. Manifolds
(Mathematics) 3. Boundary value problems.
4. FUnction spaces. I. Title. II. Series:
Lecture notes in mathematics (Berlin) ; 471.
QA3.L28 no.~71 [QA6141 510' .85 [514' .2231 75-20001
AMS Subject Classifications (1970): 35J60, 35K55, 49A20, 49F15,
53C20, 58015, 58E15, 58G99
ISBN 3-540-07185-7 Springer-Verlag Berlin' Heidelberg· New York
ISBN 0-387-07185-7 Springer-Verlag New York' Heidelberg' . Berlin
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TABLE OF CONTENTS
Foreword •• • ••••••••• 1
Part I HaI'l!loni c Map s • 3
Part II : Funct:l..on Spaces •• •• •• •• 9
•• •• • til
Part III: Semi-Elliptic and Parabolic Equations 56
Part IV The Heat Equation for Manifolds 101
Part V Growth Estimates and Convergence 125
Bibliography 166
FOREWORD
The theory of harmonic maps of manifolds has its origins in
the classic paper of Eells and Sampson [4], where existence is proved
when the target manifold has non-positive Riemannian curv~ture.
This paper generalizes this result to manifolds with boun~ary. Three
results are stated corresponding to the Dirichlet, Neumann'l!1d mi{Ced
boundary value problems. The solution to the Dirichlet problem is
proved in full detail, and we indicate the necessary minor mOdifica
tions for the other two problems at the end. The paper is divided
into five parts. and each part into sections·
Part I: Harmonic Maps
Part II: Function Spaces
Part III: Semi-Elliptic and Parabolic Equations
Part IV: The Heat Equation for Manifolds
Part V: Growth Estimates and Convergence.
In part I we define harmonic maps and state the three results. Part
II contains the definition of weighted LP spaces of potentials
and Besov spaces used in the proof and reviews their properties.
Part 111 reviews the theory of coercive linear semi-elliptic and
parabolic boundary value problems. All the material in Parts II and
III is well known to the experts in the field and can be found in
the references in tha bibliography. However. since we use several
precise and delicate facts, which are scattered over many papers
with many different definitions, and more particularly since
the exposition of the subject has profited greatly from a reoent
multiplier theorem of Stein [22], we hope the reader will find this
material a useful reference. The proof itself follows the method of
Eells and Sampson in the construction of a non-linear heat equation
for manifolds. In Part IV we prove uniqueness, regularity and
2
existence for short periods of time. This material is independent
of the curvature hypothesis, which first appears in Part V. Here
we prove some rather delicate growth estimates which guarantee that
the solution of the heat equation exists for all time and converges
to a harmonic map.
The author would especially like to express his appreciation
to James Eells Jr. for his invaluable advice and encouragement over
many years, without which this paper would never have been written;
and also to Halldor Eliasson, Ronald Goldstein, and Karen Uhlenbeck
for many helpful discussions.
Cornell Un~versity
University of Warwick
Work partially supported by the SRC and NSF.
Part I: Harmonic Maps
1. Partial differential equations for maps f:X ~ Y of one ~nl~
fold into another are of considerable interest in analysis and
topology. In this context there are no linear equations, since Y
has no additive structure. The polynomial equations of degree n
are the simplest class of equations invariant under coordinate
changes on X and Y. These are the equations given in local co-
ordinates by polynomials in derivatives of f whose degrees sum
to no more than n.
They look like
L
caf, .. . Y (f) DafD,Bf ... DYf
10.1+/.8/+. ··+IY Is.n
where the coefficients c (f) depend non-linearly on f and
a,B ••• 'V
are multi-linear functionals applied to the vectors DOf, D.Bf •..• ,DYr.
Here a denotes a multi-index (0. •...• a ) of length
1 n
10.1 = 0.1 +. .. + an and
2. The simplest ~nd most important example is Laplacels equation,
introduced for manifolds by Eells and S~~pson [4]. Let X and Y
be Riemannian manifolds with metrics and ha,B' and f:X ~ Y
a map between them. The derivative of at a point x e X 1s a
linear map Qfx: TXx ~ TYf(x) on the tangent spaces. In the
language of vector bundles, the derivative Qf is a section of the
bundle L(TX,f*TY) where f*TY is the pull-back of T¥ to a
bundle over X by the ma.p f. In local co.ordinates
Vf
4
The second derivative vVf is the derivative of Vf witp
respect to the natural connection on L(TX, f*TY). This defines
VVf ae a section of the bundle L~(TX,f*TY) of synunetric bilinear
maps. The Laplacian Af is the trace of the second derivative
VVf
Af = Tr VVf
with respect to the inner product on TX. This defines ~f as a
section of the bundle f*TY. In local coordinates, the Riemannian
connections on TX and TY are given by the Christoffel symbols
xrji ~ and yr~~~ . The pull-back connection on f*TY is given by
~ Hf3
yI'~~(f) axi .
where :1~~ (f) is yr~~ evaluated at f(x).
If E l,i.nd F are bundles. the connection on L(E,F) is given in
tensor analysis as minus the connection on E plus the connection
on F. Thus in local ~oordinates
VVf
anc;l,
Af
:;0
The map f:X + Y is ~alled harmonic if it satisfies Laplace's
eq\lation
Af = O.
Thi~ is tl'le simp:);est elljptic second orqer polynomial partial di:t'fer
ent!l,i.l equatIon for maps between manifolds.
5
3. There are many classical examples of harmonic maps.
(a) The harmonic maps X7R are the harmonic functions.
(b) The harmonic maps R~X are the geodesics.
(c) Every isometry is harmonic.
(d) A conformal map is one which preserves angles, Every conformal
map is harmonic.
(e) Every holomorphic map between ~ahler ~nifolds is harmonic.
(f) If f:X1 x X2 ~ Y is harmonic in each variable separate~y
then f is harmonic. In fact, there is a natural decomposition
£If = Alf + A2f.
(g) If G is a Lie group with a bi-invariant Riemannian metric,
then the multiplication ~:GxG ~ G is harmonic.
(h) The Hopf fibrations
are harmonic in their classical polynomial representations.
(1) If Y is Riemannian and X is a submanifold of least volume,
then the inclusion i:X ~ Y is harmonic for the induced metric on X.
4. The most important problem in the theory of harmonic maps
is to prove or disprove the following conjecture. A homotopy
class of maps of X into Y is a connected component of the
spacet1ht(x,y) of smooth maps of X into Y, wlth the COO topolOgy.
Let X and Y be compact Riemannian manifolds without boundary.
Harmonic Conjecture: There exists a harmonic map in every
homotopy class.
The best positive result is due to Eells and Sampson [4J.
Theorem. If Y has Riemannian curvature ~ 0 then there
exists a harmonic map f:X ~ Y in every homotopy class.
6
The best negative result is due to Ted Smith [21]. He con
siders harmonic maps of a sphere into an ellipsoid of revolution
which are of degree k and axially symmetric. These exist if the
ellipsoid is short and fat, but not if it is tall and thin. Thus
as the ellipsoid becomes taller and thinner, at some point the
harmonic map either bifurcates into a famiTy of axially asymmetric
maps, or it ceases to exist at all. Which happens is not known.
5. In this paper we extend the result of Eells and Sampson to
compact manifolds X and Y with boundary. There are three
natural boundary value problems.
(a) Dirichlet Problem.
We ask for a harmonic map f:X ~ Y with given values on ax.
Let h:aX ~ Y be a smooth map of ax into Y. Let~h(X,y) de
note the closed subspace of maps f:X ~ Y with flax = h. A
relative homotopy class is a connected component Of~h(X,y). If
there is a topological obstruction to extending h then~h(X,y)
is empty and nothing more can be said. Otherwise we have the
following theorem.
Theorem. Let X and Y be compact Riemannian manifolds with
boundary. Suppose that Y has Riemannian curvature ~ 0 and that
ay
is convex (or empty) . Then the Dirichlet problem for f:X ~ Y
Af 0 on X
f h on ax
has a solution in every relative homotopy class.
The condition that oy is convex is a local condition whiCh
can be expressed in terms of the Christoffel symbolS. Choose a chart
I n-l n) n L
(y , ... ,y ,y near OY such that y = (y OJ. The condition
that ay is convex is that in such a chart the matrix
I'~,B (1 ~ Cl,,B ~ n-l) is (weakly) positive definite. To see the
7
=
geometric meaning consider a geodesl.·c Ma(t) .
My ~ passl.ng throUgh a
point on ay. The equation for a geodesic says
n
If cp is tangent to OY, ~= 0 and only terms with
1 ~ a,f> ~ n-l appear. If r~~ (1 ~ a,~ ~ n-l) is positive definite
d2 n
then ~~ O. Thus the condition that OY is convex is that a
dt
geodesic tangent to oY does not enter inside Y. If X = R then
the harmonic maps are the geodesics, so the condition that o¥ is
convex is clearly necessary.
(b) Neumann Problem.
If we do not specify the map f on oX at all, we can impose
instead the auxiliary condition that the normal derivative v f = 0
'\)
on oX. Note that Vfx: TXx ~ TYf(x) and v € TXx is the normal
vector, so V'\)f = vf(v) € TYf(x) is a tangent vector on Y.
Theorem. Let X and Y be compcat -Riemannian manifolds with
boundary. Suppose that Y has Riemannian curvature ~ 0 and that
oY is convex (or empty). Then the Neumann Problem
o on X
o on oX
has a solution in every homotopy class.
(c) Mixed Problem.
The two preceding problems do not involve oY in an essential
way. This one does. Suppose that we require that f maps oX
into oY, but in an arbitrary fashion. We can then impose the
auxiliary boundary condition that the normal derivative 'lvf taken
at a point in oX should be normal to OY. This makes sense since
