Min-max minimal surfaces, horizons and electrostatic systems
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.
Cite
@article{arxiv.1912.08600,
title = {Min-max minimal surfaces, horizons and electrostatic systems},
author = {Tiarlos Cruz and Vanderson Lima and Alexandre de Sousa},
journal= {arXiv preprint arXiv:1912.08600},
year = {2025}
}
Comments
Final version. Accepted for publication in Journal of Differential Geometry. We extended one of the main results to the case where the boundary has degenerate minimal surfaces (Theorem C). Also, we now only use the Simon-Smith version of min-max theory. Finnaly, we improved the presentation of the paper, following the suggestions of the referee