  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.
