On electrostatic manifolds with boundary

TL;DR

This paper introduces electrostatic manifolds with boundary as a generalization of static manifolds, exploring their geometric properties and establishing rigidity theorems relevant to general relativity and scalar curvature problems.

Contribution

It extends the concept of static manifolds to include electric fields, analyzing the geometry of the potential's zero-level set and proving new rigidity results.

Findings

Rigidity theorems for the 3D Euclidean ball

Rigidity theorems for Reissner-Nordström manifold

Connections between zero-level set geometry and manifold properties

Abstract

Static manifolds with boundary were recently introduced by Cruz and Vit\'orio in the context of the prescribed scalar curvature problem in a manifold with boundary with prescribed mean curvature. This kind of manifold is also interesting from the point of view of the general theory of relativity. In this article, we introduce electrostatic manifolds with boundary as a natural generalization of static manifolds with boundary in the presence of a non-zero electric field. We study the geometry of the zero-level set of the potential and its connection to the global properties of electrostatic manifolds with boundary. In particular, we establish some rigidity theorems for the 3-dimensional Euclidean ball and for the Reissner-Nordstr\"om manifold.