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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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