The Cauchy problem for the Liouville equation and Bryant surfaces |
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Authors: | José A. Gá lvez,Pablo Mira |
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Affiliation: | a Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, Spain b Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, E-30203 Cartagena, Murcia, Spain |
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Abstract: | We give a construction that connects the Cauchy problem for the 2-dimensional elliptic Liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H3, and solve both of them. We construct the unique mean curvature one surface in H3 that passes through a given curve with a given unit normal along it, and provide diverse applications. In particular, topics such as period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. are treated for these surfaces. |
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Keywords: | 53A10 35J15 |
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