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On electrostatic manifolds with boundary

Stanislav Demurov, Vladimir Medvedev

TL;DR

This work introduces electrostatic manifolds with boundary as a natural extension of static manifolds by incorporating a nonzero electric field, yielding a coupled system for the static potential $V$ and the electric field $E$. It establishes critical geometric consequences, including that the boundary is totally umbilical and the zero set $\Sigma=V^{-1}(0)$ is totally geodesic, with scalar curvature $R_g=2|E|^2+2\Lambda$, and it connects these manifolds to variational principles via a functional $\mathcal{F}$. The authors develop a framework for uniqueness of non-compact electrostatic manifolds with boundary, defining quasi-local photon spheres and using RN manifolds as canonical models, and they prove rigidity theorems that classify such geometries under natural asymptotic and boundary conditions. They further relate the geometry of $\Sigma$ to global properties through a Pohozaev-type identity, deriving sharp area bounds for $\Sigma$ and clarifying when $\Sigma$ is a sphere or a disk. The appendices provide explicit Reissner–Nordström computations, including photon-sphere analysis in RN spacetimes, unifying the static and electrostatic settings within a common variational and geometric framework.

Abstract

Static manifolds with boundary were recently introduced by Cruz and Vitório 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-Nordström manifold.

On electrostatic manifolds with boundary

TL;DR

This work introduces electrostatic manifolds with boundary as a natural extension of static manifolds by incorporating a nonzero electric field, yielding a coupled system for the static potential and the electric field . It establishes critical geometric consequences, including that the boundary is totally umbilical and the zero set is totally geodesic, with scalar curvature , and it connects these manifolds to variational principles via a functional . The authors develop a framework for uniqueness of non-compact electrostatic manifolds with boundary, defining quasi-local photon spheres and using RN manifolds as canonical models, and they prove rigidity theorems that classify such geometries under natural asymptotic and boundary conditions. They further relate the geometry of to global properties through a Pohozaev-type identity, deriving sharp area bounds for and clarifying when is a sphere or a disk. The appendices provide explicit Reissner–Nordström computations, including photon-sphere analysis in RN spacetimes, unifying the static and electrostatic settings within a common variational and geometric framework.

Abstract

Static manifolds with boundary were recently introduced by Cruz and Vitório 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-Nordström manifold.
Paper Structure (13 sections, 21 theorems, 115 equations)

This paper contains 13 sections, 21 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Paper organization
  3. Acknowledgments
  4. Preliminaries
  5. General properties of electrostatic manifolds with boundary
  6. Uniqueness theorems for non-compact electrostatic manifolds with boundary
  7. Quasi-local photon spheres
  8. Asymptotically flat systems
  9. Proofs of the uniqueness theorems for non-compact electrostatic manifolds with boundary
  10. Geometry of the zero-level set of the potential
  11. Appendix
  12. Reissner--Nordström manifold
  13. Photon spheres in Reissner--Nordström manifolds

Key Result

Theorem 1.1

Let $(M^n,g,V, E)$ be a complete (up to the boundary) one-ended asymptotically Reissner--Nordström system of mass $m$ and charge $q$ (see Section sec:asymt_flat), which is an electrostatic manifold with compact boundary. Then Here $V_{m,q}$ is the Reissner--Nordström static potential in the isotropic coordinates and $E_{m,q}$ is the Reissner--Nordström electric field in the isotropic coordinates

Theorems & Definitions (52)

  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • ...and 42 more