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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 

 

The hypoelliptic Laplacian on the cotangent bundle


Author: Jean-Michel Bismut
Journal: J. Amer. Math. Soc. 18 (2005), 379-476
MSC (2000): Primary 35H10, 58A14, 58J20
Published electronically: February 28, 2005
MathSciNet review: 2137981
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Abstract: In this paper, we construct a new version of Hodge theory, where the corresponding Laplacian acts on the total space of the cotangent bundle. This Laplacian is a hypoelliptic operator, which is in general non-self-adjoint. When properly interpreted, it provides an interpolation between classical Hodge theory and the generator of the geodesic flow. The construction is also done in families in the superconnection formalism of Quillen and extends earlier work by Lott and the author.


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Additional Information

Jean-Michel Bismut
Affiliation: Département de Mathématique, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France
Email: Jean-Michel.Bismut@math.u-psud.fr

DOI: https://doi.org/10.1090/S0894-0347-05-00479-0
Keywords: Hypoelliptic equations, Hodge theory, index theory and related fixed point theorems
Received by editor(s): February 9, 2004
Published electronically: February 28, 2005
Additional Notes: The author is indebted to Viviane Baladi, Sebastian Goette and Yves Le Jan for several discussions. Gilles Lebeau’s support and enthusiasm have been essential to the whole project. A referee has also provided much help by reading the manuscript very carefully.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.