Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity

Gálvez, José; Martínez, Antonio; Milán, Francisco

doi:10.1090/S0002-9947-04-03592-5

Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity
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by José Antonio Gálvez, Antonio Martínez and Francisco Milán

Trans. Amer. Math. Soc. 356 (2004), 3405-3428

DOI: https://doi.org/10.1090/S0002-9947-04-03592-5

Published electronically: April 26, 2004

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Abstract:

In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?

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