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Transactions of the American Mathematical Society

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Embedded minimal ends of finite type


Authors: Laurent Hauswirth, Joaquín Pérez and Pascal Romon
Journal: Trans. Amer. Math. Soc. 353 (2001), 1335-1370
MSC (2000): Primary 53A10; Secondary 49Q05, 53C42
DOI: https://doi.org/10.1090/S0002-9947-00-02640-4
Published electronically: December 15, 2000
MathSciNet review: 1806738
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Abstract:

We prove that the end of a complete embedded minimal surface in $\mathbb{R} ^3$ with infinite total curvature and finite type has an explicit Weierstrass representation that only depends on a holomorphic function that vanishes at the puncture. Reciprocally, any choice of such an analytic function gives rise to a properly embedded minimal end $E$ provided that it solves the corresponding period problem. Furthermore, if the flux along the boundary vanishes, then the end is $C^0$-asymptotic to a Helicoid. We apply these results to proving that any complete embedded one-ended minimal surface of finite type and infinite total curvature is asymptotic to a Helicoid, and we characterize the Helicoid as the only simply connected complete embedded minimal surface of finite type in $\mathbb{R} ^3$.


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Additional Information

Laurent Hauswirth
Affiliation: Department of Mathematics, University of Fortaleza, 60811-341 Fortaleza, Brazil
Address at time of publication: Equipe d’Analyse et de Mathematiques Appliquees, Universite de Marne-la-Vallee, 2 rue de la Butte Verte, 93166 Noisy-le-Grand Cedex, France
Email: hauswirth@math.univ-mlv.fr

Joaquín Pérez
Affiliation: Departamento de Geometria y Topologia, Universidad de Granada, Fuentenueva s/n, 18071, Granada, Spain
Email: jperez@goliat.ugr.es

Pascal Romon
Affiliation: Equipe d’Analyse et de Mathematiques Appliquees, Universite de Marne-la-Vallee, 2 rue de la Butte Verte, 93166 Noisy-le-Grand Cedex, France
Email: romon@math.univ-mlv.fr

DOI: https://doi.org/10.1090/S0002-9947-00-02640-4
Keywords: Minimal surface, finite type, Helicoid
Received by editor(s): March 8, 1999
Received by editor(s) in revised form: September 29, 1999
Published electronically: December 15, 2000
Additional Notes: The research of the second author was partially supported by a DGYCYT Grant No. PB97-0785.
Article copyright: © Copyright 2000 American Mathematical Society

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