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 electrostatic 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. Combining this with a theorem by Marques and Neves we obtain a uniqueness result for electrostatic systems. We also prove an inequality relating the area and the charge of a minimal surface of index one in a Cauchy data satisfying the Dominant Energy Condition.