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We present a connection between minimal surfaces of index one and GeneralRelativity. 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-maxproblem for the area functional, in particular it has index one. We also obtainan inequality relating the area and the charge of a minimal surface of indexone in a Cauchy data satisfying the Dominant Energy Condition fornon-electromagnetic matter fields. Moreover, we explore a global version ofthis inequality, and the rigidity in the case of the equality, using a resultproved by Marques and Neves.

Min-max minimal surfaces, horizons and electrostatic systems