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Min-max minimal surfaces, horizons and electrostatic systems

Tiarlos Cruz, Vanderson Lima, Alexandre de Sousa

TL;DR

This work establishes a tight link between index-one minimal surfaces and electrostatic general relativity. By developing a min-max framework for unstable horizons in complete electrostatic systems under the DEC, it shows such horizons arise as index-one area-maximizing surfaces, and it derives a sharp area–charge inequality that ties geometric and electromagnetic data. The authors use Min-Max theory (à la Marques–Neves) and Plateau-type arguments to prove rigidity results: global classifications occur when equality is attained, with conclusions matching standard models like the Reissner–Nordström–de Sitter and de Sitter spaces. The paper further analyzes a rich catalog of exact electrostatic models, assesses the horizon indices, and derives topological consequences for allowed Cauchy data, contributing to the classification program for electrostatic systems and their horizons.

Abstract

We present a connection between minimal surfaces of index one and General Relativity. First, we show that for a certain class of (electro)static systems, each of its unstable horizons is the solution of a one-parameter min-max problem for the area functional, in particular it has index one. We also obtain an inequality relating the area and the charge of a minimal surface of index one in a Cauchy data satisfying the Dominant Energy Condition for non-electromagnetic matter fields. Moreover, we explore a global version of this inequality, and the rigidity in the case of the equality, using a result proved by Marques and Neves.

Min-max minimal surfaces, horizons and electrostatic systems

TL;DR

This work establishes a tight link between index-one minimal surfaces and electrostatic general relativity. By developing a min-max framework for unstable horizons in complete electrostatic systems under the DEC, it shows such horizons arise as index-one area-maximizing surfaces, and it derives a sharp area–charge inequality that ties geometric and electromagnetic data. The authors use Min-Max theory (à la Marques–Neves) and Plateau-type arguments to prove rigidity results: global classifications occur when equality is attained, with conclusions matching standard models like the Reissner–Nordström–de Sitter and de Sitter spaces. The paper further analyzes a rich catalog of exact electrostatic models, assesses the horizon indices, and derives topological consequences for allowed Cauchy data, contributing to the classification program for electrostatic systems and their horizons.

Abstract

We present a connection between minimal surfaces of index one and General Relativity. First, we show that for a certain class of (electro)static systems, each of its unstable horizons is the solution of a one-parameter min-max problem for the area functional, in particular it has index one. We also obtain an inequality relating the area and the charge of a minimal surface of index one in a Cauchy data satisfying the Dominant Energy Condition for non-electromagnetic matter fields. Moreover, we explore a global version of this inequality, and the rigidity in the case of the equality, using a result proved by Marques and Neves.

Paper Structure

This paper contains 23 sections, 17 theorems, 169 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Electrostatic systems
  3. Standard Static Electrovacuum Space-times
  4. General properties of electrostatic systems
  5. Standard Models
  6. List of models
  7. The Reissner-Nordström solution
  8. The de Sitter solution
  9. The (charged) Nariai solution
  10. A quotient of the Nariai system
  11. The cold black hole solution
  12. The ultra cold black hole solution
  13. The index of the horizons in the models
  14. Topology of electrostatic systems
  15. An area-decreasing flow
  16. ...and 8 more sections

Key Result

Lemma 4

Consider $(M^3,g)$ be a Riemannian manifold, $E \in \mathfrak{X}(M)$ and $V\in C^{\infty}(M)$ satisfying the following system of equations: for some constant $\Lambda \in \mathbb{R}$. Suppose $V$ is not identically zero and ${\Sigma=V^{-1}(0)}$ is nonempty, then:

Figures (2)

  • Figure 1: Standard models. The plot is of the dimensionless quantities $m^2\Lambda$ vs. $Q^2\Lambda$.
  • Figure 2: A sweepout by spheres. The point in red belongs to the singular set $P$.

Theorems & Definitions (39)

  • Definition 1
  • Remark 2
  • Remark 3
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Definition 6
  • Proposition 7
  • proof
  • ...and 29 more