### The hypoelliptic Laplacian and Ray-Singer metrics, Jean-Michel Bismut, Gilles Lebeau

Creator

Contributor

Summary

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

Language

eng

Literary Form

non fiction

Edition

Course Book

Note

Description based upon print version of record

Physical Description

1 online resource (378 p.)

Specific Material Designation

remote

Form Of Item

online

Isbn

9786612458378

### Subject

- Matrix calculus
- Classical Wiener space
- Fredholm determinant
- Fiber bundle
- Logarithm
- Integration by parts
- Wave equation
- Parametrix
- Eigenvalues and eigenvectors
- Differential operator
- Cohomology
- Theory
- Computation
- Riemann–Roch theorem
- Laplacian operator
- Direct proof
- Smoothness
- Hodge theory
- Coordinate system
- Resolvent set
- Dirac operator
- Mellin transform
- Asymptotic expansion
- Martingale (probability theory)
- Self-adjoint operator
- Trace class
- Square root
- Variable (mathematics)
- Parameter
- Eigenform
- Bijection
- Ground state
- Estimation
- Determinant
- Symmetric space
- Hilbert space
- Principal bundle
- Translational symmetry
- Brownian motion
- Connection form
- Malliavin calculus
- Embedding
- Self-adjoint
- Parity (mathematics)
- Alexander Grothendieck
- Rectangle
- Holomorphic function
- Fourier transform
- Vector bundle
- Euclidean space
- Scientific notation
- Commutator
- Summation
- Morse theory
- Chaos theory
- Torus
- Curvature
- Probabilistic method
- Theorem
- Combination
- Equation
- Sign convention
- Differentiable manifold
- Fokker–Planck equation
- Holomorphic vector bundle
- Explicit formulae (L-function)
- Heat kernel
- Explicit formula
- Formal power series
- Cotangent bundle
- Polynomial
- Notation
- Covariance matrix
- Hypoelliptic operator
- Function space
- Ellipse
- Feynman–Kac formula
- Derivative
- Differential equations, Hypoelliptic
- Clifford algebra
- Uniform convergence
- Metric spaces
- Girsanov theorem
- Tangent space
- Asymptote
- Transversality (mathematics)
- Brownian dynamics
- Ricci curvature
- Taylor series
- Fourier series
- Spectral theory
- De Rham cohomology
- Invertible matrix
- Chern class
- Berezin integral
- Curvature tensor
- Sobolev space
- Analytic function
- Projection (linear algebra)
- Stochastic calculus
- Supertrace
- Vector space
- Stochastic process