Self-adjointness of the Gaffney Laplacian on vector bundles

TL;DR

This paper investigates the self-adjointness of the Gaffney Laplacian on vector bundles over possibly incomplete Riemannian manifolds, establishing conditions related to the boundary's nature.

Contribution

It proves the self-adjointness of the Gaffney Laplacian under the polar boundary condition and characterizes it via negligible boundary, advancing understanding of geometric analysis on incomplete manifolds.

Findings

Self-adjointness holds if the Cauchy boundary is polar.

Negligible boundary is necessary and sufficient for self-adjointness.

Provides criteria for self-adjointness on incomplete manifolds.

Abstract

We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator.