Minimal isometric immersions into S^2 x R and H^2 x R

TL;DR

This paper investigates minimal isometric immersions of simply connected surfaces into S^2 x R and H^2 x R, establishing conditions for their classification and the nature of their continuous families.

Contribution

It relates the problem to PDE systems and classifies constant curvature minimal surfaces, also describing the structure of continuous families of immersions.

Findings

Constant curvature minimal surfaces are either totally geodesic or associate to catenoids.

Continuous families of minimal immersions are all associate surfaces.

The problem reduces to solving specific PDE systems on the surface.

Abstract

For a given simply connected Riemannian surface Sigma, we relate the problem of finding minimal isometric immersions of Sigma into S^2 x R or H^2 x R to a system of two partial differential equations on Sigma. We prove that a constant intrinsic curvature minimal surface in S^2 x R or H^2 x R is either totally geodesic or part of an associate surface of a certain limit of catenoids in H^2 x R. We also prove that if a non constant curvature Riemannian surface admits a continuous one-parameter family of minimal isometric immersions into S^2 x R or H^2 x R, then all these immersions are associate.