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Applications of Partial Differential Equations to Problems in Geometry

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Applications of Partial Differential Equations To Problems in Geometry Jerry L. Kazdan Preliminary revised version Copyright c 1983, 1993 by Jerry L. Kazdan ° Preface These notes are from an intensive one week series of twenty lectures given to a mixed audience of advanced graduate students and more experienced mathematicians in Japan in July, 1983. As a consequence, these they are not aimed at experts, and are frequently quite detailed, especially in Chapter 6 where a variety of standard techniques are presented. My goal was to in- troduce geometers to some of the techniques of partial differential equations, and to introduce those working in partial differential equations to some fas- cinating applications containing many unresolved nonlinear problems arising in geometry. My intention is that after reading these notes someone will feel that they can cope with current research articles. In fact, the quite sketchy Chapter 5 and Chapter 6 are merely intended to be advertisements to read the complete details in the literature. When writing something like this, there is the very real danger that the only people who understand anything are those who already know the subject. Caveat emptor. In any case, I hope I have shown that if one assumes a few basic results on Sobolev spaces and elliptic operators, then the basic techniques used in the applications are comprehensible. Of course carrying out the details for any specific problem may be quite complicated—but at least the ideas should be clearly recognizable. These notes definitely do not represent the whole subject. I did not have time to discuss a number of beautiful applications such as minimal surfaces, harmonic maps, global isometric embeddings (including the Weyl and Minkowski problems as well as Nash’s theorem), Yang-Mills fields, the wave equation and spectrum of the Laplacian, and problems on compact manifolds with boundary or complete non-compact manifolds. In addition, these lectures discuss only existence and uniqueness theorems, and ignore other more qualitative problems. Although existence results seem to hold the center of the stage in contemporary applications, a more balanced discussion would be important in a longer series of lectures. The lectures assumed some acquaintance with either Riemannian geom- etry or partial differential equations. While mathematicians outside of these areas should be able to follow these notes, it may be more difficult for them to appreciate the significance of the questions or results. By the ruthless schedule of my charming hosts, these notes are to be typed shortly after the completion of the lectures. My hosts felt (wisely, I think) that it would be more useful to have an informal set of lecture notes available quickly rather than with longer time for a more polished manuscript. Inevitably, as befits a first draft, there will be rough edges and outright errors. I hope none of these are serious and would appreciate any corrections and suggestions for subsequent versions. One thing I know I would do is add a few additional sections to Chapter i 1. In particular, there should really be some mention of Green’s functions and at least a vague summary of the story for boundary value problems— especially the Dirichlet problem (see [N-3], pp. 41-50 for what I have in mind). Also, the dry, technical flavor of Chapter 1 should be balanced by a few more easy—but useful—applications of the linear theory. For instance, Moser’s result on volume forms [MJ-1] uses only simple Hodge theory. But my time deadline has come. I hope these notes are useful to someone seeking a rapid introduction with a minimum of background. This task is made much easier because of the recent books [Au-4] and [GT], where one can find most of the missing details. I am grateful to many Japanese mathematicians. In addition to helping make my visit so pleasant, they are also proofreading the typed manuscript; all I’ll see is the finished product. Finally, I wish to give special thanks to Professor T. Ochiai for his extraordinary hospitality and thoughtfulness. I also thank the National Science Foundation for their support. Srinagar, India 10 August 1983 Note added, June, 1993. This is an essentially unrevised version of the lectures I gave in Japan in July, 1983. The only notable addition is a section discussing the Hodge Theorem, I also took advantage of the retyping into TEX to make a few corrections and minor clarifications in the wording. Alas, retyping introduces its own errors. [To Do: incorporate the following into the preface] Throughout these lectures we will need some background material on elliptic and, to a lesser extent, parabolic partial differential operators. Equa- tions that are neither elliptic nor parabolic do arise in geometry (a good example is the equation used by Nash to prove isometric embedding results); however many of the applications involve only elliptic or parabolic equations. For this material I have simply inserted a slightly modified version of an Ap- pendix I wrote for the book [Be-2]. This book may also be consulted for basic formulas in geometry.2 At some places, I have added supplementary information that will be used later in the lectures. I suggest that one should skim this chapter quickly, paying more attention to the examples than to the generalities, and then move directly to Chapter 6. One can refer back to the introductory material if the need arises. Most of our treatment is restricted to compact manifolds without bound- ary. This is simply to avoid the extra steps required to adequately discuss 2For reference, some basic geometry formulas are collected in an Appendix at the end of these notes. ii appropriate boundary conditions. One can also eliminate most of the com- plications in thinking about manifolds by restricting attention to the two dimensional torus with its Euclidean metric, so the Laplacian is the basic uxx + uyy , and one is considering only doubly periodic functions, say with period 2π . Even simpler, yet still often fruitful and non-trivial, is to reduce to the one dimensional case of functions on the circle. Here ∆u = +u′′ . This also points out one critical sign convention: for us the Laplacian has the sign so that ∆u =+u′′ for functions on R1 (except that in the special case of the Hodge Laplacian on differential forms, we write ∆ = dd∗ + d∗d as in equation (2.4) below, where in the particular case of 0 -forms this gives the opposite sign). To discuss the Laplacian and related elliptic differential operators, one must introduce certain function spaces. It turns out that the spaces one thinks of first, namely C0,C1,C2 , etc. are, for better or worse, not ap- propriate; one is forced to use more complicated spaces. For instance, if ∆u = f Ck , one would like to have u Ck+2 . With the exception of the special one∈ dimensional case covered by∈ the theory of ordinary differential equations, this is false for these Ck spaces (see the example in [Mo, p. 54]), but which is true for the spaces to be introduced now. For proofs and more details see [F, 8-11] and [GT]. Unless stated§ otherwise, to be safe we will always assume that the open sets we consider are connected. For simplicity M will always denote a C∞ connected Riemannian mani- fold without boundary, n = dim M , and E and F are smooth vector bundles (with inner products) over M . Of course, there are related assertions if M has a boundary or if M is not C∞ . Sometimes we will write (M n,g) if we wish to point out the dimension and the metric, g . The volume element is ∞ written dxg , or sometimes dx . By smooth we always mean C ; we write Cω for the space of real analytic functions. We also use standard multi-index notation, so if x = (x1,...,xn) is a n point in R and j = (j1,...,jn) is a vector of non-negative integers, then j j1 jn j j1 jn j = j1 + + jn , x = x x , and ∂ =(∂/∂x1) (∂/∂xn) | | ··· 1 ··· n ··· —————— Here and below we will use the notation a(x, ∂ku) , F (x, ∂ku) , etc. to represent any (possibly nonlinear) differential operator of order k (so here ∂ku actually represents the k-jet of u ). Last Revised: February 29, 2016 iii iv Contents Preface i 1 Linear Differential Operators 1 1.1 Introduction.................................... 1 1.2 H¨older Spaces . 6 1.3 Sobolev Spaces . 7 1.4 Sobolev Embedding Theorem . 8 1.5 Adjoint ...................................... 10 1.6 PrincipalSymbol ................................. 11 2 Linear Elliptic Operators 13 2.1 Introduction.................................... 13 2.2 TheDefinition .................................. 14 2.3 Schauder and Lp Estimates........................... 16 2.4 Regularity (smoothness) . 17 2.5 Existence ..................................... 18 2.6 TheMaximumPrinciple............................. 21 2.7 Proving the Index Theorem . 25 2.8 LinearParabolicEquations ........................... 26 3 Geometric Applications 29 3.1 Introduction.................................... 29 3.2 HodgeTheory................................... 29 a) HodgeDecomposition ............................ 29 b) Poincar´eDuality ............................... 30 c) The de Rham Complex . 31 3.3 Eigenvalues of the Laplacian . 32 3.4 Bochner Vanishing Theorems . 36 a) One-parameter Isometry Groups . 36 b) Harmonic 1 forms ............................. 38 − 3.5 TheDiracOperator ............................... 39 3.6 The Lichnerowicz Vanishing Theorem . 40 3.7 ALiouvilleTheorem ............................... 41 3.8 UniqueContinuation............................... 43 a) TheQuestion................................. 43 4 Nonlinear Elliptic Operators 45 4.1 Introduction.................................... 45 4.2 DifferentialOperators .............................. 45 4.3 Ellipticity ..................................... 46 v 4.4 Nonlinear Elliptic Equations: Regularity . 47 4.5 Nonlinear Elliptic Equations: Existence . 48 4.6 AComparisonTheorem ............................. 50 4.7 Nonlinear Parabolic Equations . 52 4.8 AListofTechniques ............................... 52 5 Examples of Techniques 55 5.1 Introduction.................................... 55 5.2 CalculusofVariations .............................. 57 5.3 ContinuityMethod................................ 60 5.4 Schauder Fixed Point Theorem .
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