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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.

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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