
The Hypoelliptic Laplacian and Ray-Singer Metrics
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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.
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Content
- Frontmatter,
- Contents,
- Introduction,
- Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles,
- Chapter 2. The hypoelliptic Laplacian on the cotangent bundle,
- Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel,
- Chapter 4. Hypoelliptic Laplacians and odd Chern forms,
- Chapter 5. The limit as t ¿ +8 and b ¿ 0 of the superconnection forms,
- Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics,
- Chapter 7. The hypoelliptic torsion forms of a vector bundle,
- Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula,
- Chapter 9. A comparison formula for the Ray-Singer metrics,
- Chapter 10. The harmonic forms for b ¿ 0 and the formal Hodge theorem,
- Chapter 11. A proof of equation (8.4.6),
- Chapter 12. A proof of equation (8.4.8),
- Chapter 13. A proof of equation (8.4.7),
- Chapter 14. The integration by parts formula,
- Chapter 15. The hypoelliptic estimates,
- Chapter 16. Harmonic oscillator and the J0 function,
- Chapter 17. The limit of A'2fb,±H as b ¿ 0,
- Bibliography,
- Subject Index,
- Index of Notation,
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