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

TL;DR

This paper studies minimal surfaces in the product of two 2-spheres with a neutral metric, classifies totally geodesic surfaces, relates minimal surfaces to Gordon equations, and provides topological classifications of compact cases.

Contribution

It offers a comprehensive classification of minimal and totally geodesic surfaces in these neutral metric spaces, linking them to Gordon equations and topology.

Findings

All totally geodesic surfaces are computed.

Minimal surfaces are related to solutions of Gordon equations.

Topological classification of compact minimal surfaces is provided.

Abstract

We investigate minimal surfaces in products of two-spheres ${\mathbb S}^2_p\times {\mathbb S}^2_p$, with the neutral metric given by $(g,-g)$. Here ${\mathbb S}^2_p\subset {\mathbb R}^{p,3-p}$ , 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.