Lecture 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 This work is' subject to copyright. 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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