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 $\mathbf{S}^2_p\times\mathbf{S}^2_p$, with the neutral metric given by $(g,-g)$. Here $\mathbf{S}^2_p\subset \mathbf{R}^{p,3-p}$, 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.