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

Francisco Torralbo, Francisco Urbano

Published 2008-07-11, updated 2008-10-16Version 2

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.