Surfaces with parallel mean curvature vector in $\mathbb {S}^2\times \mathbb {S}^2$ and $\mathbb {H}^2\times \mathbb {H}^2$

Abstract:

Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in $\mathbb {S}^2\times \mathbb {S}^2$ and $\mathbb {H}^2\times \mathbb {H}^2$ are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of $\mathbb {S}^2\times \mathbb {R}$ and $\mathbb {H}^2\times \mathbb {R}$ is established. Using this, 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 $\mathbb {S}^2\times \mathbb {R}$ and $\mathbb {H}^2\times \mathbb {R}$ is proved.