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](http://books.google.com/books/about/The_Laplacian_on_a_Riemannian_Manifold.html?id=gzJ6Vn0y7XQC)). 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](http://projecteuclid.org/euclid.cmp/1103921772), [Osgood-Philips-Sarnak](http://www.sciencedirect.com/science/article/pii/0022123688900705), [Okikiolu](http://arxiv.org/abs/math/0103239)).


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](http://projecteuclid.org/euclid.jdg/1214439284) 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](http://projecteuclid.org/euclid.mmj/1029003949)). 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?



Heat kernel 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](http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1988_4_21_3/ASENS_1988_4_21_3_307_0/ASENS_1988_4_21_3_307_0.pdf)


[Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale](http://archive.numdam.org/ARCHIVE/AIF/AIF_1989__39_1/AIF_1989__39_1_73_0/AIF_1989__39_1_73_0.pdf)


[Spectral Asymptotics for Operators of Hormander Type](http://arxiv.org/abs/1310.8649)


More explicit computations can be done in contact geometry


[Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry](http://arxiv.org/abs/1105.1285)


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


[The Subelliptic Heat Kernel on the CR sphere](http://arxiv.org/abs/1112.3084)


[Trace of heat kernel, spectral zeta function and isospectral problem for sub-laplacians](http://link.springer.com/article/10.1007%2Fs11425-009-0186-4)


