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Generalized Hopf differentials. (English) Zbl 1118.53036
The authors extend the idea of the Hopf differential in the study of constant mean curvature surfaces of the sphere to surfaces in product spaces like $$S^2\times \mathbb R$$ and $$H^2\times \mathbb R$$ by restoring holomorphicity by a geometric correction term. More generally this construction can be applied to surfaces in homogeneous 3-manifolds that fibre over a surface with totally geodesic fibres, and this is the appropriate class for which the developed method will work.

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C20 Global Riemannian geometry, including pinching 53C30 Differential geometry of homogeneous manifolds