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A Hopf differential for constant mean curvature surfaces in $$\mathbb S^2 \times \mathbb R$$ and $$\mathbb H^2 \times\mathbb R$$. (English) Zbl 1078.53053
A well-known theorem by H. Hopf states, that any immersed constant mean curvature (cmc) sphere $$S^2\to\mathbb{R}^3$$ is in fact a standard distance sphere with radius $$1/H$$, where $$H$$ is this cmc. Here, the key is the discovery that the complexification of the traceless part of the second fundamental form $$h_\Sigma$$ of an immersed surface $$\Sigma^2$$ with constant $$H$$ in $$\mathbb{R}^3$$ is a holomorphic quadratic differential $$Q$$ on $$\Sigma^2$$. Afterwards this theorem has been extended replacing $$\mathbb{R}^3$$ by $$\mathbb{S}^3$$ or $$\mathbb{H}^3$$. This differential $$Q$$ played also a significant role in the discovery of the Wente tori and in the subsequent development of a general theory of cmc tori in $$\mathbb{R}^3$$.
In the present paper a generalized quadratic differential $$Q$$ is introduced for $$\Sigma^2$$ in $$\mathbb{S}^2\times\mathbb{R}$$ and $$\mathbb{H}^2\times\mathbb{R}$$ (in other words, in $$M^2_\kappa\times\mathbb{R}$$, where $$M^2_\kappa$$ is of constant curvature $$\kappa$$). Hopf’s result is extended to cmc spheres in these target spaces. It is proved that such immersed spheres are surfaces of revolution. Four distinct classes of complete, possibly immersed, cmc surfaces $$\Sigma^2\to M^2_\kappa\times\mathbb{R}$$ with vanishing quadratic differential $$Q$$ are found. They are also geometrically described in detail. For $$\kappa< 0$$ all four cases do actually occur, but if $$\kappa\geq 0$$ then only the embedded cmc spheres $$S^2_H\subset M_\kappa\times\mathbb{R}$$ occur.

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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