Rigidity of the subelliptic heat kernel on $\operatorname{SU}(2)$

TL;DR

This paper investigates the rigidity properties of the subelliptic heat kernel on the Lie group SU(2), showing that certain heat kernel structures uniquely determine the Hopf fibration and the sub-Riemannian sphere.

Contribution

It proves that a heat kernel of a specific form on SU(2) characterizes the Hopf fibration and the sub-Riemannian sphere, establishing a rigidity result.

Findings

Heat kernel of special form implies bundle-isometry to Hopf fibration

Characterization of the sub-Riemannian sphere via heat kernel rigidity

Uniqueness of the heat kernel structure on SU(2)

Abstract

We study heat kernel rigidity for the Lie group $\operatorname{SU}\left( 2 \right)$ kernel equipped with a sub-Riemannian structure. We prove that a metric measure space equipped with a heat kernel of a special form is bundle-isometric to the Hopf fibration $\operatorname{U}\left( 1 \right)\to \operatorname{SU}\left( 2 \right)\to \mathbb{CP}^1$, which coincides with the sub-Riemannian sphere $\operatorname{SU}\left( 2 \right)$.