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A basic tool in the theory of constant mean curvature (cmc) surfaces Σ in space forms is the holomorphic quadratic differential discovered by H. Hopf. In this paper we generalize this differential to immersed cmc surfaces Σ in the product spaces S × R and H × R and prove a corresponding result about the geometry of immersed cmc spheres S in these target spaces.

A Hopf differential for constant mean curvature surfaces in S2×R and H2×R