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Linear Analysis on Manifolds: Notes for Math 7376, Spring 2016

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Linear Analysis on Manifolds: Notes for Math 7376, Spring 2016 Linear Analysis on Manifolds: Notes for Math 7376, Spring 2016 Chris Kottke June 14, 2016 2 Linear Analysis on Manifolds Introduction These notes were written to accompany a graduate class taught at Northeastern University in Spring 2016. The goal was to cover some classical topics concerning linear elliptic operators on compact manifolds, including elliptic regularity and Fredholm theory, spectral asymptotics (Weyl's formula), and the local Atiyah-Singer index formula using heat kernel methods. The point of view of these notes is decidedly microlocal, in the sense that operators are studied in terms of their (distributional) Schwartz kernels. Objects of interest, such as gen- eralized inverses of elliptic operators and heat kernels, are first approximated by constructing parametrices, which are then improved by some iterative procedure and then compared to the true objects in order to deduce important properties of the latter. The audience for the class was mixed, with some students having prior expertise with pseudodifferential operators, and other students having limited analytical background. For this reason, the somewhat unusual choice was made to use pseudodifferential operators on manifolds in order to prove key results about elliptic operators, but to skip the technical development of these operators. Thus we take an axiomatic approach, positing the existence of a class of operators satisfying a handful of fundamental axioms, which constitute a kind of user's interface for pseudodifferential operators. There are many good sources for the rigorous development of pseudodifferential operators on manifolds. Among these I mention in particular Pierre Albin's excellent notes [Alb15] written for a similar course at UIUC, which include a detailed background on distribution theory, the requisite Riemannian geometry, and a rather complete development of ΨDOs, in addition to the topics covered here. I followed Albin's approach quite closely in places, and was under the impression that I was complementing his work by covering the Atiyah-Singer theorem in these notes, which was not covered in detail in the first version [Alb12] of notes. It was only after the end of my course that I discovered Albin's later version [Alb15] of his notes, updated to include the index theorem. Thus it is probably the case that these notes constitute a proper subset of Albin's notes, though I hope some readers may yet benefit from the different exposition here, however slightly it may differ! Chris Kottke, June 14, 2016. 3 Contents Introduction 3 1 Elliptic theory on compact manifolds 7 1.1 Differential operators . .7 1.1.1 Principal symbols . .9 1.1.2 Bundles . 11 1.1.3 Adjoints . 13 1.2 Pseudodifferential operators . 16 1.2.1 Distributions on compact Riemannian manifolds . 17 1.2.2 Pseudodifferential operators . 21 1.3 Ellipticity . 24 1.3.1 Parametrices . 25 1.3.2 Elliptic regularity . 27 1.3.3 Fredholm property of elliptic operators . 27 1.4 Hodge Theory . 32 1.4.1 Distributional Hodge theory . 35 1.4.2 Elliptic complexes . 36 1.5 L2, Sobolev spaces, and spectral theory . 37 1.5.1 L2 mapping properties . 37 1.5.2 Unbounded operators and closed extensions . 40 1.5.3 Sobolev spaces . 41 1.5.4 Spectral theory . 46 2 Spectral theory and heat kernels 51 2.1 Overview . 51 2.2 Heat kernel of Laplacian on Euclidean space . 53 2.3 Heat kernel on a manifold . 56 2.3.1 Blow-up . 56 2.3.2 Heat space . 57 2.3.3 Kernels . 59 2.3.4 Action of differential operators . 61 2.3.5 Heat Parametrix . 62 4 Contents 5 2.3.6 Parametrix for Laplace-type operators . 63 2.3.7 True solution . 65 2.4 Trace class operators . 68 2.4.1 Integral kernels . 71 2.5 Heat Trace and Weyl asymptotics . 73 3 Atiyah-Singer index theorem 77 3.1 Overview . 77 3.2 Dirac operators . 80 3.2.1 Clifford algebras and representations . 80 3.2.2 Dirac operators on a manifold . 85 3.2.3 Spin . 90 3.2.4 Spin structures on a manifold . 93 3.2.5 Curvature and Bochner formulas . 96 3.2.6 Supertrace . 102 3.3 Heat kernels and Getzler rescaling . 104 3.3.1 Rescaling a bundle at a hypersurface . 105 3.3.2 Getzler rescaling . 108 3.3.3 Mehler's formula . 117 3.4 The index theorem . 119 3.4.1 Non-spin manifolds . 120 3.4.2 A bit of Chern-Weil theory . 121 3.4.3 Applications . 124 Bibliography 127 6 Linear Analysis on Manifolds Chapter 1 Elliptic theory on compact manifolds 1.1 Differential operators n n Consider R with coordinates x = (x1; : : : ; xn). A differential operator on R is a linear 1 n operator on C (R ) of the form X α P u = aα(x)@x u(x): (1.1) jα|≤k 1 n Here the coefficients aα(x) 2 C (R ) are smooth functions and we we employ multi-index n P α notation, where α = (α1; : : : ; αn) 2 N , jαj = i αi, and @x is shorthand for the mixed partial derivative operator @α1 @αn α α1 αn @x = @x1 ··· @xn = ··· : @x1 @xn The integer k 2 N is the order of the operator. Suppose now M is a smooth manifold of dimension n. Recall that this means M has a maximal atlas of smoothly compatible coordinate charts (U; U 0; φ), where =∼ 0 n φ : U ⊂ M ! U ⊂ R is a homeomorphism, and \smoothly compatible" means that −1 0 n 0 n φb ◦ φa : Ua \ φa(Ub) ⊂ R ! Ub \ φb(Ua) ⊂ R is a diffeomorphism. We will typically omit φ and U 0 from the notation, and observe the convention of regarding the coordinate functions xi := xi ◦ φ : U ! R as being functions on U ⊂ M itself; thus we will say x = (x1; : : : ; xn) are local coordinates on U ⊂ M. Definition 1.1. A differential operator of order k on M is a linear operator on C1(M) given locally by expressions of the form (1.1). In other words, on any coordinate chart U, the function P u restricted to U has the form (1.1). 7 8 Linear Analysis on Manifolds Observe that this is well-defined; namely, if we have a smooth change of coordinates x = 0 0 −1 0 x(x ), then (@x1 ;:::;@xn ) = @x = D(x; x ) @x where 0 h @xj i D(x; x ) = 0 @xi is the Jacobian matrix, whose entries are smooth functions. It follows that (1.1) becomes 0 X 0 0 α 0 P u(x ) = aα(x )@x0 u(x ) jα|≤k 0 0 0 for a new set of coefficients aα(x ). The general expression for the aα in terms of the aα is quite complicated! However, we will soon see that the top order part behaves nicely. We denote by Diffk(M) the set of differential operators of order at most k on M. It is easy to see that this is a vector space over R (or C if we use complex valued functions), and that for all l ≤ k, we have inclusions Diffl(M) ⊂ Diffk(M): In particular Diff0(M) = C1(M) is nothing more than the smooth functions on M, considered as multiplication operators on C1(M). Diff1(M) includes Diff0(M) as well as the smooth vector fields V(M) = C1(M; TM), which we recall is the (vector) space of linear derivations on C1(M): V(M) 3 V : C1(M) ! C1(M);V (fg) = f V (g) + g V (f); and these have local coordinate expressions V = a1(x)@x1 + ··· + an(x)@xn : Of course, as operators on C1(M), we may compose differential operators, and it is easy to see that Diffk(M) ◦ Diffl(M) ⊂ Diffk+l(M): (1.2) Again, we may verify this in local coordinates, but observe that if X α X β P = aα(x)@x ;Q = bβ(x)@x ; jα|≤k jβ|≤l X γ P ◦ Q = cγ(x)@x ; jγ|≤k+l then the general formulas for cγ in terms of the aα and bβ are complicated! Algebraically speaking, the set [ Diff(M) = Diffk(M) k2N of all differential operators has the structure of an associative filtered algebra1, where the filtration is by N. The term ‘filtered’ here simply reflects the fact that Diff(M) is a union of subsets indexed by N, and (1.2) holds. 1Some prefer the term `ring' here, which is certainly applicable, though Diff(M) is also a vector space over R (or C, if we allow complex coefficients), so we will prefer the term `algebra'. Chapter 1. Elliptic theory on compact manifolds 9 1.1.1 Principal symbols The next order of business is to show that the highest order terms of differential operators behave nicely. Let us revert to the Euclidean setting for just a moment. Definition 1.2. Let X α P = aα(x)@x jα|≤k n be a differential operator of order k on R . The principal symbol of P is the (complex-valued) 1 n n 1 ∗ n function σk(P ) 2 C (R × R ) = C (T R ) given by k X α σk(P )(x; ξ) = i aα(x)ξ ; (1.3) jαj=k α α1 αn where ξ = (ξ1; : : : ; ξn) and ξ = ξ1 ··· ξn . Note that the sum is only over terms of order exactly k. We will often omit the subscript k from the notation and just write σ(P ) := σk(P ). Observe that not only is σ(P ) smooth in both variables, it is in fact a (homogeneous) polynomial of order k in the ξ variables. The factor of i, which is a standard convention, seems a bit annoying at this point, but it would cause much more pain later on if we leave it off.
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