Minimal surfaces in the product of two dimensional real space forms endowed with a neutral metric

Abstract

We investigate minimal surfaces in products of two-spheres ${\mathbb{S}}_p^2\times {\mathbb{S}}_p^2$, with the neutral metric given by $(g,-g)$. Here ${\mathbb{S}}_p^2\subset {\mathbb{R}}^{p,3-p}$ , and $g$ is the induced metric on the sphere. We compute all totally geodesic surfaces and we give a relation between minimal surfaces and the solutions of the Gordon equations. Finally, in some cases we give a topological classification of compact minimal surfaces.