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 | References | Similar Articles | Additional Information

We prove that the end of a complete embedded minimal surface in 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 provided that it solves the corresponding period problem. Furthermore, if the flux along the boundary vanishes, then the end is -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 .

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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