# Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold

Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. Rosenberg). The heat operator trace is interesting because it can be used to prove the meromorphic continuation to $\mathbb{C}$ of the spectral zeta function $$\zeta(\Delta, g,s) = \sum_{\lambda_i\neq 0}\lambda_i^{-s}.$$ The meromorphic continuation of $\zeta$ is regular at zero; hence if $\Delta$ has a trivial kernel, we can differentiate in the complex parameter to obtain the zeta-regularized determinant $$\det\Delta = \exp(-\zeta'(\Delta,g,0)).$$

Over the years, the search for extrema for instances of $\det\Delta$ has attracted a lot of attention (Onofri, Osgood-Philips-Sarnak, Okikiolu).

I'd like to know how much progress has been made with respect to defining regularized determinants for other "Laplace-like" operators. To do so, the development of heat kernel asymptotics would seem to be a necessary first step.

For example, Beals, Greiner, and Stanton showed that the heat operator trace of the sublaplacian on a CR manifold has an asymptotic expansion analogous to that for the Laplacian in the Riemannian case. The heat operator trace of the pseudoconformal Laplacian on a strongly pseudoconvex CR manifold also has an asymptotic expansion (cf. Stanton, 1989). Have spectral zeta functions been defined for these operators? Do they have interesting invariance properties?

More to the point, do we have similar asymptotics for the sublaplacian on a contact Riemannian manifold?

A lot is known about asymptotics of subelliptic heat kernels.

There are Minakshinsudaram-Pleijel expansions that hold for very general subelliptic operators and they apply in particular to the contact case:

Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus

Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale

Spectral Asymptotics for Operators of Hormander Type

More explicit computations can be done in contact geometry

Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry

and on some model spaces like $S^{2n+1}$

The Subelliptic Heat Kernel on the CR sphere

Trace of heat kernel, spectral zeta function and isospectral problem for sub-laplacians

I would take a look at Noncommutative Microlocal Analysis, Memoirs AMS 313, available here.

This will probably not answer all your questions, but particularly in reference to your last question: take a look at Chapter III, Section 5, and also the calculation that starts from (4.47) of page 71 (the analysis on the Heisenberg group is done here).