Heat flow out of a compact manifold

TL;DR

This paper investigates the asymptotic behavior of heat content in a compact manifold without boundary, revealing boundary-localized terms and deriving geometric expressions for initial asymptotic coefficients.

Contribution

It introduces a method to analyze heat content asymptotics without boundary conditions using pseudo-differential calculus and invariance theory.

Findings

Established existence of complete asymptotic series for heat content

Derived initial terms of the asymptotic expansion in terms of geometric data

Identified boundary-localized terms in the asymptotics

Abstract

We discuss the heat content asymptotics associated with the heat flow out of a smooth compact manifold in a larger compact Riemannian manifold. Although there are no boundary conditions, the corresponding heat content asymptotics involve terms localized on the boundary. The classical pseudo-differential calculus is used to establish the existence of the complete asymptotic series and methods of invariance theory are used to determine the first few terms in the asymptotic series in terms of geometric data. The operator driving the heat process is assumed to be an operator of Laplace type.