Sub-Riemannian Limit of the Differential Form Heat Kernels of Contact Manifolds

We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the η-invariant and the determinant of the Laplacian. In particular, we prove that contact versions of the relative η-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological.