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Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature. II. (English) Zbl 1058.53008
This paper is a continuation of a preceding work of the same authors with the same title. In the present paper they classify all complete surfaces of constant mean curvature 1 in hyperbolic 3-space \(H^3\) which have total absolute curvature not exceeding \(4\pi\). As the authors point out, the total curvature of such surfaces need not be an integer multiple of \(4\pi\). The classification given is explicit in terms of the holomorphic data of the surface, namely the secondary Gauss map and the Hopf differential. Here the secondary Gauss map of a constant mean curvature 1 surface \(f: M\to H^3\) is a local isometry from \((\widetilde M, -Kds^2)\) into \(\mathbb{C}\mathbb{P}^1\) endowed with the Fubini-Study metric, where \(\widetilde M\) denotes the universal cover of \(M\) and \(ds^2\) and \(K\) are the induced metric and the Gauss curvature of \(f\), respectively.
For part I see W. Rossman, M. Umehara and K. Yamada [Hiroshima Math. J. 34, No. 1, 21–56 (2004; Zbl 1088.53004)].

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A35 Non-Euclidean differential geometry
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Zbl 1088.53004
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