On electrostatic manifolds with boundary

Abstract

Static manifolds with boundary were recently introduced in the context of the problem of prescribing both the scalar curvature of a Riemannian manifold and the mean curvature of its boundary. These manifolds are also of interest from the perspective of general relativity. In this article, we introduce electrostatic manifolds with boundary as a natural generalization of static manifolds with boundary in the presence of a nonzero electric field. We also introduce Robin electrostatic quadruples, extending the notion of Robin static triples previously introduced by the second author. We establish several uniqueness theorems for suitable regions of the Reissner-Nordström manifold. Finally, we characterize the criticality of the first eigenvalue of an associated mixed Robin-Dirichlet Schrödinger operator under admissible volume-preserving variations in terms of the equality of the surface gravities along the Dirichlet boundary components.

arXiv comment32 pages; comments are welcome!