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

TL;DR

This paper investigates the small time behavior of the heat kernel in 3D contact sub-Riemannian geometry, explicitly computing key coefficients in terms of geometric invariants.

Contribution

It provides explicit formulas for the first two coefficients of the heat kernel asymptotics in 3D contact structures, linking them to fundamental invariants.

Findings

Explicit formulas for heat kernel coefficients in 3D contact geometry

Connection between heat kernel asymptotics and geometric invariants

Perturbative approach to small time heat kernel analysis

Abstract

In this paper we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We then explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in terms of the two basic functional invariants $\chi$ and $\kappa$ defined on a 3D contact structure.