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.
Frontmatter, pg. iContents, pg. vIntroduction, pg. 1Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles, pg. 11Chapter 2. The hypoelliptic Laplacian on the cotangent bundle, pg. 25Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel, pg. 44Chapter 4. Hypoelliptic Laplacians and odd Chern forms, pg. 62Chapter 5. The limit as t â +â and b â 0 of the superconnection forms, pg. 98Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics, pg. 113Chapter 7. The hypoelliptic torsion forms of a vector bundle, pg. 131Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula, pg. 162Chapter 9. A comparison formula for the Ray-Singer metrics, pg. 171Chapter 10. The harmonic forms for b â 0 and the formal Hodge theorem, pg. 173Chapter 11. A proof of equation (8.4.6), pg. 182Chapter 12. A proof of equation (8.4.8), pg. 190Chapter 13. A proof of equation (8.4.7), pg. 194Chapter 14. The integration by parts formula, pg. 214Chapter 15. The hypoelliptic estimates, pg. 224Chapter 16. Harmonic oscillator and the J0 function, pg. 247Chapter 17. The limit of A'2Ï b,+/-H as b â 0, pg. 264Bibliography, pg. 353Subject Index, pg. 359Index of Notation, pg. 361
Series: Annals of Mathematics Studies (Paperback)
Tertiary; University or College
Number Of Pages: 367
Published: 7th September 2008
Country of Publication: US
Dimensions (cm): 23.11 x 15.24
Weight (kg): 0.54