The Hypoelliptic Laplacian and Ray-Singer Metrics. (AM-167) (Annals of Mathematics Studies, 167)

This book presents the analytic foundations to the theory of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator acting on the cotangent bundle of a compact manifold, is supposed to interpolate between the classical Laplacian and the geodesic flow. Jean-Michel Bismut and Gilles Lebeau establish the basic functional analytic properties of this operator, which is also studied from the perspective of local index theory and analytic torsion. The book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. The authors give the proper functional analytic setting in order to study this operator and develop a pseudodifferential calculus, which provides estimates on the hypoelliptic Laplacian's resolvent. When the deformation parameter tends to zero, the hypoelliptic Laplacian converges to the standard Hodge Laplacian of the base by a collapsing argument in which the fibers of the cotangent bundle collapse to a point. For the local index theory, small time asymptotics for the supertrace of the associated heat kernel are obtained. The Ray-Singer analytic torsion of the hypoelliptic Laplacian as well as the associated Ray-Singer metrics on the determinant of the cohomology are studied in an equivariant setting, resulting in a key comparison formula between the elliptic and hypoelliptic analytic torsions. Contents Introduction Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles 1.1 The Clifford algebra 1.2 The standard Hodge theory 1.3 The Levi-Civita superconnection 1.4 Superconnections and Poincaré duality 1.5 A group action 1.6 The Lefschetz formula 1.7 The Riemann-Roch-Grothendieck theorem 1.8 The elliptic analytic torsion forms 1.9 The Chern analytic torsion forms 1.10 Analytic torsion forms and Poincaré duality 1.11 The secondary classes for two metrics 1.12 Determinant bundle and Ray-Singer metric Chapter 2. The hypoelliptic Laplacian on the cotangent bundle 2.1 A deformation of Hodge theory 2.2 The hypoelliptic Weitzenböck formulas 2.3 Hypoelliptic Laplacian and standard Laplacian 2.4 A deformation of Hodge theory in families 2.5 Weitzenböck formulas for the curvature and Levi-Civita superconnection 2.7 The superconnection and Poincaré duality 2.8 A 2-parameter rescaling 2.9 A group action Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel 3.1 The cohomology of T*X and the Thom isomorphism 3.2 The Hodge theory of the hypoelliptic Laplacian 3.3 The heat kernel for 3.4 Uniform convergence of the heat kernel as b → 0 3.5 The spectrum of AS b → 0 3.6 The Hodge condition 3.7 The hypoelliptic curvature Chapter 4. Hypoelliptic Laplacians and odd Chern forms 4.1 The Berezin integral 4.2 The even Chern forms 4.3 The odd Chern forms and a 1-form on R*2 4.4 The limit as t → 0 of the forms 4.5 A fundamental identity 4.6 A rescaling along the fibers of T*X 4.7 Localization of the problem 4.8 Replacing and the rescaling of Clifford variables on T*X 4.9 The limit as t → 0 of the rescaled operator 4.10 The limit of the rescaled heat kernel 4.11 Evaluation of the heat kernel for [omitted] 4.12 An evaluation of certain supertraces 4.13 A proof of Theorems 4.2.1 and 4.4.1 Chapter 5. The limit as t → + ∞ and b → 0 of the superconnection forms 5.1 The definition of the limit forms 5.2 The convergence results 5.3 A contour integral 5.4 A proof of Theorem 5.3.1 5.5 A proof of Theorem 5.3.2 5.6 A proof of the first equations in (5.2.1) and (5.2.2) Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics 6.1 The hypoelliptic torsion forms 6.2 Hypoelliptic torsion forms and Poincaré duality 6.3 A generalized Ray-Singer metric on the determinant of the cohomology 6.4 Truncation of the spectrum and Ray-Singer metrics 6.5 A smooth generalized metric on the determinant bundle 6.6 The equivariant determinant 6.7 A variation formula 6.8 A simple identity 6.9 The projected connections 6.10 A proof of Theorem 6.7.2 Chapter 7. The hypoelliptic torsion forms of a vector bundle 7.1 The function τ (c, η, x) 7.2 Hypoelliptic curvature for a vector bundle 7.3 Translation invariance of the curvature 7.4 An automorphism of E 7.5 The von Neumann supertrace of exp 7.6 A probabilistic expression for 7.7 Finite dimensional supertraces and infinite determinants 7.8 The evaluation of the form 7.9 Some extra computations 7.10 The Mellin transform of certain Fourier series 7.11 The hypoelliptic torsion forms for vector bundles Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula 8.1 On some secondary Chern classes 8.2 The main result 8.3 A contour integral 8.4 Four intermediate results 8.5 The asymptotics of the 8.6 Matching the divergences 8.7 A proof of Theorem 8.2.1 Chapter 9. A comparison formula for the Ray-Singer metrics Chapter 10. The harmonic forms for b → 0 and the formal Hodge theorem 10.1 A proof of Theorem 8.4.2 10.2 The kernel of [omitted] as a formal power series 10.3 A proof of the formal Hodge Theorem 10.4 Taylor expansion of harmonic forms near b = 0 Chapter 11. A proof of equation (8.4.6) 11.1 The limit of the rescaled operator as t → 0 11.2 The limit of the supertrace as t → 0 11.3 A proof of equation (8.4.6) Chapter 12. A proof of equation (8.4.8) 12.1 Uniform rescalings and trivializations 12.2 A proof of (8.4.8) Chapter 13. A proof of equation (8.4.7) 13.1 The estimate in the range 13.2 Localization of the estimate near 13.3 A uniform rescaling on the creation annihilation operators 13.4 The limit as t → 0 of the rescaled operator 13.5 Replacing X by Tx X 13.6 A proof of (13.2.11) 13.7 A proof of Theorem 13.6.2 Chapter 14. The integration by parts formula 14.1 The case of Brownian motion 14.2 The hypoelliptic diffusion 14.3 Estimates on the heat kernel 14.4 The gradient of the heat kernel Chapter 15. The hypoelliptic estimates 15.1 The operator 15.2 A Littlewood-Paley decomposition 15.3 Projectivization of T*X and Sobolev spaces 15.4 The hypoelliptic estimates 15.5 The resolvent on the real line 15.6 The resolvent on C 15.7 Trace class properties of the resolvent Chapter 16. Harmonic oscillator and the J0 function 16.1 Fock spaces and the Bargman transform 16.2 The operator B(ξ) 16.3 The spectrum of B(i ξ) 16.4 The function J0 (y, λ) 16.5 The resolvent of B(iξ) + P Chapter 17. The limit of as b → 0 17.1 Preliminaries in linear algebra 17.2 A matrix expression for the resolvent 17.3 The semiclassical Poisson bracket 17.4 The semiclassical Sobolev spaces 17.5 Uniform hypoelliptic estimates for P[sub(h)] 17.6 THE OPERATOR AND ITS RESOLVENT Sh,λ FOR λ ∈ R 17.7 The resolvent THE RESOLVENT Sh,λ FOR λ ∈ C 17.8 A trivialization over X and the symbols 17.9 The symbol and its inverse 17.10 The parametrix for Sh,λ 17.11 A localization property for 17.12 The operator 17.13 A proof of equation (17.12.9) 17.14 An extension of the parametrix to λ ∈ V 17.15 Pseudodifferential estimates for 17.16 The operator Θh,λ 17.17 The operator Th,λ 17.18 The operator (J1/J0) 17.19 The operator Uh,λ 17.20 Estimates on the resolvent of Th,h2λ 17.21 The asymptotics of (Lc − λ) −1 17.22 A localization property Bibliography Subject Index Index of Notation A B C D E F G H I J K L M N O P Q R S T U V W X Y