The Hypoelliptic Laplacian and Ray-Singer Metrics
In this book, analytic theory of the hypoelliptic Laplacian is developed and corresponding results on the associated Ray-Singer torsion are established. The hypoelliptic Laplacian is a second order differential operator defined on the cotangent bundle of a compact Riemannian manifold. It is supposed to interpolate the classical Laplacian and an operator related to a geodesic flow. In this way, it gives a semiclassical version of the fact that the Witten Laplacian on the corresponding loop space should interpolate between the classical Hodge Laplacian and Morse theory for the same energy functional. The authors develop Hodge theory for the studied Laplacian and the local index theory of the associated heat kernel. They adapt the theory of Ray-Singer torsion and the analytic torsion forms of Bismut-Lott and they develop an appropriate pseudodifferential calculus. They show that when the deformation parameter tends to zero, the hypoelliptic Laplacian tends to the usual Hodge Laplacian of the base space by a collapsing argument letting the fibre shrink to a point. They also obtain small time asymptotics for the supertrace of the associated heat kernel. A comparison formula between the elliptic and hypoelliptic analytic torsions is derived, studying an equivariant setting of the Ray-Singer torsion of the studied operator and the associated Ray-Singer metrics on the determinant of the cohomology ring. To obtain localisation estimates, probabilistic methods related to diffusion processes are used.