Self-adjointness of the Gaffney Laplacian on vector bundles
  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. 
Lashi Bandara and Ognjen Milatovic
math.FA math.DG math.SP