Heat kernel asymptotics for quaternionic contact manifolds

TL;DR

This paper computes the small-time asymptotics of the heat kernel on quaternionic contact manifolds, revealing how the coefficients relate to geometric quantities like scalar curvature, and applies these results to spectral invariance in the subriemannian context.

Contribution

It explicitly calculates the first two heat kernel coefficients for quaternionic contact manifolds and links the second coefficient to the qc scalar curvature, advancing understanding of subriemannian spectral geometry.

Findings

The first heat kernel coefficient $c_0$ is computed explicitly.

The second coefficient $c_1$ depends linearly on the qc scalar curvature.

Spectral invariance of geometric quantities is established for compact qc-Einstein manifolds.

Abstract

In this paper, we study the heat kernel associated to the intrinsic sublaplacian on a quaternionic contact manifold considered as a subriemannian manifold. More precisely, we explicitly compute the first two coefficients $c_0$ and $c_1$ appearing in the small time asymptotics expansion of the heat kernel on the diagonal. We show that the second coefficient $c_1$ depends linearly on the qc scalar curvature $\kappa$. Finally we apply our results to compact qc-Einstein manifolds and prove the spectral invariance of geometric quantities in the subriemannian setting.