On the heat diffusion for generic Riemannian and sub-Riemannian structures

TL;DR

This paper analyzes the small-time heat kernel asymptotics near the cut locus for various generic low-dimensional Riemannian and sub-Riemannian structures, revealing the nature of singularities and their impact on heat diffusion.

Contribution

It provides the first detailed asymptotic analysis at the cut locus for generic 3D contact and 4D quasi-contact sub-Riemannian manifolds, and classifies singularities of the exponential map in low-dimensional Riemannian cases.

Findings

Only $A_3$ and $A_5$ singularities can occur along minimizing geodesics in generic low-dimensional Riemannian manifolds.

Explicit small-time heat kernel asymptotics are derived near the cut locus for the studied structures.

Non-generic cases exhibit a wide variety of asymptotic behaviors, even on Riemannian surfaces.

Abstract

In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As a byproduct, we show that, for generic low-dimensional Riemannian manifolds, the only singularities of the exponential map, as a Lagragian map, that can arise along a minimizing geodesic are $A_3$ and $A_5$ (in the classification of Arnol'd's school). We show that in the non-generic case, a cornucopia of asymptotics can occur, even for Riemannian surfaces.