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.
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