The subelliptic heat kernel on the anti-de Sitter spaces

TL;DR

This paper derives explicit formulas for the subelliptic heat kernel and sub-Riemannian distance on anti-de Sitter spaces, providing insights into their geometric and analytical properties.

Contribution

It presents the first explicit, geometrically meaningful formulas for the subelliptic heat kernel and sub-Riemannian distance on anti-de Sitter spaces, leveraging symmetry from the Hopf fibration.

Findings

Explicit formula for the subelliptic heat kernel

Small time asymptotics of the heat kernel

Explicit sub-Riemannian distance formula

Abstract

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space H2n+1 which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. The key idea is to work in a set of coordinates that reflects the symmetry coming from the Hopf fibration S1->H2n+1. A direct application is obtaining small time asymptotics of the subelliptic heat kernel. Also we derive an explicit formula for the sub-Riemannian distance on H2n+1