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New minimal surfaces in the hyperbolic space. (English) Zbl 1390.53057
Summary: We obtain new complete minimal surfaces in the hyperbolic space \(\mathbb{H}^3\), by using Ribaucour transformations. Starting with the family of spherical catenoids in \(\mathbb{H}^3\) found by H. Mori [Indiana Univ. Math. J. 30, 787–794 (1981; Zbl 0589.53007)], we obtain 2- and 3-parameter families of new minimal surfaces in the hyperbolic space, by solving a non trivial integro-differential system. Special choices of the parameters provide minimal surfaces whose parametrizations are defined on connected regions of \(\mathbb{R}^{2}\) minus a disjoint union of Jordan curves. Any connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of \(\mathbb{H}^3\) is a closed curve. The geometric properties of the surfaces regarding the ends, completeness and symmetries are discussed.
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
Full Text: DOI
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