The Subelliptic Heat Kernel on the CR sphere

TL;DR

This paper derives explicit formulas for the heat kernel, Green function, and sub-Riemannian distance on the CR sphere, leveraging symmetries from the sphere's fibration to advance understanding of subelliptic operators.

Contribution

It provides a geometrically meaningful explicit formula for the subelliptic heat kernel on the CR sphere, recovering known Green functions and distances through symmetry-adapted coordinates.

Findings

Explicit heat kernel formula derived

Green function of conformal sub-Laplacian recovered

Sub-Riemannian distance formula obtained

Abstract

We study the heat kernel of the sub-Laplacian L on the CR sphere S2n+1. An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub- Laplacian -L + n2 that was obtained by Geller [12], and also get an explicit formula for the sub-Riemannian distance. The key point is to work in a set of coordinates that reflects the symmetries coming from the fibration S2n+1 \rightarrow CPn.