The relative heat content for submanifolds in sub-Riemannian geometry

TL;DR

This paper investigates the small-time behavior of the relative heat content for submanifolds in sub-Riemannian geometry, establishing foundational results and highlighting limitations of approximation methods.

Contribution

It proves the existence of smooth tubular neighborhoods for non-characteristic submanifolds and introduces a definition and approximation of relative heat content in this setting.

Findings

Smooth tubular neighborhoods exist for submanifolds without characteristic points.

The proposed approximation does not fully capture the asymptotic expansion.

Explicit example demonstrates the failure of the approximation method.

Abstract

We study the small-time asymptotics of the relative heat content for submanifolds in sub-Riemannian geometry. First, we prove the existence of a smooth tubular neighborhood for submanifolds of any codimension, assuming they do not have characteristic points. Next, we propose a definition of relative heat content for submanifolds of codimension $k\geq 1$ and we build an approximation of this quantity, via smooth tubular neighborhoods. Finally, we show that this approximation fails to recover the asymptotic expansion of the relative heat content of the submanifold, by studying an explicit example.