Transactions of the American Mathematical Society

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

Author(s): José Antonio Gálvez; Antonio Martínez; Francisco Milán
Journal: Trans. Amer. Math. Soc. 356 (2004), 3405-3428.
MSC (2000): Primary 53C42; Secondary 53A35
Posted: 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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José Antonio Gálvez
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: jagalvez@ugr.es

Antonio Martínez
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: amartine@ugr.es

Francisco Milán
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: milan@ugr.es

DOI: 10.1090/S0002-9947-04-03592-5
PII: S 0002-9947(04)03592-5
Keywords: Hyperbolic 3-space, Weingarten surfaces, Plateau problem, Weierstrass data.
Received by editor(s): November 11, 2002
Posted: April 26, 2004
Additional Notes: This research was partially supported by MCYT-FEDER Grant No. BFM2001-3318 and Junta de Andalucía CEC: FQM0804
Copyright of article: Copyright 2004, American Mathematical Society

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