Applications of Partial Differential Equations To Problems ...

1 Applications of Partial Differential EquationsTo Problems in GeometryJerry L. KazdanPreliminary revised versionCopyrightc 1983, 1993 by Jerry L. KazdanPrefaceThese notes are from an intensive one week series of twenty lectures givento a mixed audience of advanced graduate students and more experiencedmathematicians in Japan in July, 1983. As a consequence, these they are notaimed at experts, and are frequently quite detailed, especially in Chapter 6where 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 arisingin geometry. My intention is that after reading these notes someone will feelthat they can cope with current research articles.

2 In fact, thequite sketchyChapter 5 and Chapter 6 are merely intended to be advertisements to readthe complete details in the literature. When writing something like this,there is the very real danger that the only people who understand anythingare those who already know the any case, I hope I have shown that if one assumes a few basic resultsonSobolev spaces and elliptic operators, then the basic techniques used in theapplications are comprehensible. Of course carrying out the details for anyspecific problem may be quite complicated but at least the ideasshould beclearly notes definitely do not represent the whole subject. I did nothave time to discuss a number of beautiful Applications such as minimalsurfaces, harmonic maps, global isometric embeddings (including the Weyland Minkowski Problems as well as Nash s theorem), Yang-Mills fields, thewave equation and spectrum of the Laplacian, and Problems on compactmanifolds with boundary or complete non-compact addition,these lectures discuss only existence and uniqueness theorems, and ignoreother more qualitative Problems .

3 Although existence results seem to hold thecenter of the stage in contemporary Applications , a more balanced discussionwould be important in a longer series of lectures assumed some acquaintance with either Riemanniangeom-etry or Partial Differential Equations . While mathematiciansoutside of theseareas should be able to follow these notes, it may be more difficult for themto appreciate the significance of the questions or the ruthless schedule of my charming hosts, these notes are to betyped shortly after the completion of the lectures. My hosts felt (wisely, Ithink) that it would be more useful to have an informal set of lecture notesavailable quickly rather than with longer time for a more polished , as befits a first draft, there will be rough edges and outright hope none of these are serious and would appreciate any corrections andsuggestions for subsequent thing I know I would do is add a few additional sections to Chapteri1.

4 In particular, there should really be some mention of Green s functionsand at least a vague summary of the story for boundary value Problems especially the Dirichlet problem (see [N-3], pp. 41-50 for whatI have inmind). Also, the dry, technical flavor of Chapter 1 should be balanced by afew more easy but useful Applications of the linear theory. For instance,Moser s result on volume forms [MJ-1] uses only simple Hodge theory. Butmy time deadline has hope these notes are useful to someone seeking a rapid introductionwith a minimum of background. This task is made much easier because of therecent books [Au-4] and [GT], where one can find most of the missing am grateful to many Japanese mathematicians. In addition to helping makemy visit so pleasant, they are also proofreading the typed manuscript; all I llsee is the finished product.

5 Finally, I wish to give special thanks to ProfessorT. Ochiai for his extraordinary hospitality and thoughtfulness. I also thankthe National Science Foundation for their , India10 August 1983 Note added, June, is an essentially unrevised version of thelectures I gave in Japan in July, 1983. The only notable addition is a sectiondiscussing the Hodge Theorem, I also took advantage of the retyping intoTEX 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 onelliptic and, to a lesser extent, parabolic Partial Differential operators. Equa-tions that are neither elliptic nor parabolic do arise in geometry (a goodexample is the equation used by Nash to prove isometric embeddingresults);however many of the Applications involve only elliptic or parabolic this material I have simply inserted a slightly modified version of an Ap-pendix I wrote for the book [Be-2].

6 This book may also be consulted forbasic formulas in some places, I have added supplementaryinformation that will be used later in the lectures. I suggest that one shouldskim this chapter quickly, paying more attention to the examples than to thegeneralities, and then move directly to Chapter 6. One can refer back to theintroductory material if the need of our treatment is restricted to compact manifolds without bound-ary. This is simply to avoid the extra steps required to adequately discuss2 For reference, some basic geometry formulas are collected in an Appendix at the endof these boundary conditions. One can also eliminate most of the com-plications in thinking about manifolds by restricting attention to the twodimensional torus with its Euclidean metric, so the Laplacianis the basicuxx+uyy, and one is considering only doubly periodic functions, say withperiod 2.

7 Even simpler, yet still often fruitful and non-trivial, is toreduceto the one dimensional case of functions on the circle. Here u= +u .This also points out one critical sign convention: for us the Laplacian hasthe sign so that u= +u for functions onR1(except that in the specialcase of the Hodge Laplacian on Differential forms, we write =dd +d das in equation ( ) below, where in the particular case of0-forms this givestheoppositesign).To discuss the Laplacian and related elliptic Differential operators, onemust introduce certain function spaces. It turns out that the spaces onethinks of first, namelyC0, 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 haveu Ck+2. With the exception of thespecial one dimensional case covered by the theory of ordinary differentialequations, this isfalsefor theseCkspaces (see the example in [Mo, p.)]

8 54]),but which is true for the spaces to be introduced now. For proofs and moredetails see [F, 8-11] and [GT].Unless stated otherwise, to be safe we will always assume that the opensets we consider are simplicityMwill always denote aC connected Riemannian mani-fold without boundary,n= dimM, andEandFare smooth vector bundles(with inner products) overM. Of course, there are related assertions ifMhas a boundary or ifMis notC . Sometimes we will write (Mn, g) if wewish to point out the dimension and the metric,g. The volume element iswrittendxg, or sometimesdx. Bysmoothwe always meanC ; we writeC for the space of real analytic also use standard multi-index notation, so ifx= (x1, .. , xn) is apoint inRnandj= (j1, .. , jn) is a vector of non-negative integers, then|j|=j1+ +jn,xj=xj11 xjnn, and j= ( / x1)j1 ( / xn)jn Here and below we will use the notationa(x, ku) ,F(x, ku) , etc.

9 Torepresent any (possibly nonlinear) Differential operator of orderk(so here kuactually represents thek-jet ofu).Last Revised: February 29, 2016iiiivContentsPrefacei1 Linear Differential Introduction .. H older Spaces .. Sobolev Spaces .. Sobolev Embedding Theorem .. Adjoint .. Principal Symbol .. 112 Linear Elliptic Introduction .. The Definition .. Schauder andLpEstimates .. Regularity (smoothness) .. Existence .. The Maximum Principle .. Proving the Index Theorem .. Linear Parabolic Equations .. 263 Geometric Introduction .. Hodge Theory .. 29a) Hodge Decomposition .. 29b) Poincar e Duality .. 30c) The de Rham Complex .. Eigenvalues of the Laplacian .. Bochner Vanishing Theorems .. 36a) One-parameter Isometry Groups .. 36b) Harmonic 1 forms.

10 The Dirac Operator .. The Lichnerowicz Vanishing Theorem .. A Liouville Theorem .. Unique Continuation .. 43a) The Question .. 434 Nonlinear Elliptic Introduction .. Differential Operators .. Ellipticity .. Nonlinear Elliptic Equations : Regularity .. Nonlinear Elliptic Equations : Existence .. A Comparison Theorem .. Nonlinear Parabolic Equations .. A List of Techniques .. 525 Examples of Introduction .. Calculus of Variations .. Continuity Method .. Schauder Fixed Point Theorem .. Sub and Supersolutions .. The Heat Equation .. Summary for u=f(x) k(x)eu.. 696 Implicit Function Introduction .. Isothermal Coordinates .. Complex Structures .. 72a) Complex Structures onR2.. 72b) Complex Structures onR2n.. Prescribing Gauss and Scalar Curvature.