The Hypoelliptic Laplacian and Ray Singer Metrics AM 167 1st edition by Jean Michel Bismut ISBN 0691137323 9780691137322

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ISBN 10: 0691137323

ISBN 13: 9780691137322

Author:  Jean Michel Bismut

Table of contents: 

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
2.6 [omitted][sup(M)][sub(Φb,±H–bωH)], [omitted][sup(M,2)][sub(Φb,±H–b&
2.7 The superconnection A[sup(M)][sub(Φ,H–ωH)] 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 [omitted][sup(2)][sub(Φ,Hc)]3.4 Uniform convergence of the heat kernel as b → 0
3.5 The spectrum of [omitted][sup(‘2)][sub(Φb,±H)] 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*[sup(2)]4.4 The limit as t → 0 of the forms u[sub(b,t)], v[sub(b,t)], w[sub(b,t)]4.5 A fundamental identity
4.6 A rescaling along the fibers of T*X
4.7 Localization of the problem
4.8 Replacing T* X by T[sub(x)]X [omitted] T*[sub(x)]X and the rescaling of Clifford variables on T*
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 (–[omitted][sup(E)][sub(c)] )
7.6 A probabilistic expression for Q’[sub(c)]7.7 Finite dimensional supertraces and infinite determinants
7.8 The evaluation of the form Tr[sub(s)] [g exp (–[omitted][sup(E)][sub(c)] )]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 I[sup(0)][sub(k)]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 t ≤ b[sup(β)]13.2 Localization of the estimate near π[sup(–1)] X[sub(g)]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 T[sub(x)]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 [omitted][sup(‘2)][sub(Φb, ±H)]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 J[sub(0)] function
16.1 Fock spaces and the Bargman transform
16.2 The operator B(ξ)
16.3 The spectrum of B(iξ)
16.4 The function J[sub(0)] (y, λ)
16.5 The resolvent of B(iξ) + P
Chapter 17. The limit of [omitted][sup(‘2)][sub(Φb, ±H)] 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 P[sup(0)][sub(h)] and its resolvent S[sub(h,λ)] for λ ∈ R
17.7 The resolvent S[sub(h,λ)] for λ ∈ C
17.8 A trivialization over X and the symbols S[sup(d,k)][sub(ρ,δ,c)]17.9 The symbol Q[sup(h)][sub(0)] (x,ξ) – λ and its inverse e[sub(0,h,λ)] (x, &#958
17.10 The parametrix for S[sub(h,λ)]17.11 A localization property for E[sub(0)], E[sub(1)]17.12 The operator P[sub(±)]S[sub(h,λ)]17.13 A proof of equation (17.12.9)
17.14 An extension of the parametrix to λ ∈ [omitted]17.15 Pseudodifferential estimates for P[sub(±)]S[sub(h,λ)] i±
17.16 The operator Θ[sub(h,λ)]17.17 The operator T[sub(h,λ)]17.18 The operator (J[sub(1)]/J[sub(0)]) (hD[sup(X)]/√2, λ)
17.19 The operator U[sub(h,λ)]17.20 Estimates on the resolvent of T[sub(h,h[sup(2)]λ]17.21 The asymptotics of (L[sub(c)] – λ)[sup(–1)]17.22 A localization property

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Jean Michel Bismut,Laplacian,Hypoelliptic