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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.
MSC:
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
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