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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Properness of associated minimal surfaces


Authors: Antonio Alarcón and Francisco J. López
Journal: Trans. Amer. Math. Soc. 366 (2014), 5139-5154
MSC (2010): Primary 53A10; Secondary 32H02, 53C42
Published electronically: June 16, 2014
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Abstract: For any open Riemann surface $ \mathcal {N}$ and finite subset $ \mathfrak{Z}\subset \mathbb{S}^1=\{z\in \mathbb{C}\,\vert\;\vert z\vert=1\},$ there exist an infinite closed set $ \mathfrak{Z}_{\mathcal {N}} \subset \mathbb{S}^1$ containing $ \mathfrak{Z}$ and a null holomorphic curve $ F=(F_j)_{j=1,2,3}:\mathcal {N}\to \mathbb{C}^3$ such that the map

$\displaystyle \mathfrak{Y}:\mathfrak{Z}_{\mathcal {N}}\times \mathcal {N} \to \... ...ad \mathfrak{Y}(\mathfrak{v},P)={\rm Re}\big (\mathfrak{v} (F_1,F_2)(P)\big ), $

is proper.

In particular, $ {\rm Re}\big (\mathfrak{v} F\big ):\mathcal {N} \to \mathbb{R}^3$ is a proper conformal minimal immersion properly projecting into $ \mathbb{R}^2\equiv \mathbb{R}^2\times \{0\} \subset \mathbb{R}^3,$ for all $ \mathfrak{v} \in \mathfrak{Z}_{\mathcal {N}}.$


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Additional Information

Antonio Alarcón
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, Spain
Email: alarcon@ugr.es

Francisco J. López
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, Spain
Email: fjlopez@ugr.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06050-9
PII: S 0002-9947(2014)06050-9
Keywords: Null holomorphic curves, associated minimal surfaces
Received by editor(s): March 7, 2012
Published electronically: June 16, 2014
Additional Notes: The first author was supported by Vicerrectorado de Política Científica e Investigación de la Universidad de Granada and was partially supported by MCYT-FEDER grants MTM2007-61775 and MTM2011-22547, Junta de Andalucía Grant P09-FQM-5088, and the grant PYR-2012-3 CEI BioTIC GENIL (CEB09-0010) of the MICINN CEI Program
The second author was partially supported by MCYT-FEDER research projects MTM2007-61775 and MTM2011-22547, and Junta de Andalucía Grant P09-FQM-5088
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.