Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2

TL;DR

This paper constructs holomorphic differentials for surfaces with parallel mean curvature in product spaces, establishes a correspondence with constant mean curvature surfaces, classifies special cases, and proves rigidity results.

Contribution

It introduces holomorphic Hopf differentials for these surfaces, links them to constant mean curvature surfaces, and classifies spheres with parallel mean curvature vector.

Findings

Constructed two holomorphic Hopf differentials.

Established a 1:1 correspondence with pairs of constant mean curvature surfaces.

Classified surfaces with vanishing Hopf differentials, including spheres.

Abstract

Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in S^2xS^2 and H^2xH^2 are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of S^2xR and H^2xR is established. Using that, surfaces with vanishing Hopf differentials (in particular spheres with parallel mean curvature vector) are classified and a rigidity result for constant mean curvature surfaces of S^2xR and H^2xR is proved.