Heat conservation for generalized Dirac Laplacians on manifolds with boundary

TL;DR

This paper establishes a heat conservation principle for generalized Dirac Laplacians on manifolds with boundary, extending previous results and providing new insights into harmonic sections and applications to geometric operators.

Contribution

It proves a heat conservation principle under mixed boundary conditions for generalized Dirac Laplacians, extending previous results and applying a domination property via a Feynman-Kac formula.

Findings

Conservation principle holds under geometric controls and boundary conditions.

Extension of Feynman-Kac formula to this setting.

Vanishing results for $L^2$ harmonic sections.

Abstract

We consider a notion of conservation for the heat semigroup associated to a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitable way and imposing uniform lower bounds on the zero order (Weitzenb\"ock) piece of the Dirac Laplacian and on the endomorphism defining the mixed boundary condition we show that the corresponding conservation principle holds. A key ingredient in the proof is a domination property for the heat semigroup which follows from an extension to this setting of a Feynman-Kac formula recently proved in \cite{dL1} in the context of differential forms. When applied to the Hodge Laplacian acting on differential forms satisfying absolute boundary conditions, this extends previous results by Vesentini \cite{Ve} and Masamune \cite{M} in the boundaryless case. Along the way we also prove a vanishing result for $L^2$ harmonic sections in the broader context of generalized (not necessarily Dirac) Laplacians. These results are further illustrated with applications to the Dirac Laplacian acting on spinors and to the Jacobi operator acting on sections of the normal bundle of a free boundary minimal immersion.