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Transactions of the American Mathematical Society

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Compact complete minimal immersions in $ \mathbb{R}^3$


Author: Antonio Alarcón
Journal: Trans. Amer. Math. Soc. 362 (2010), 4063-4076
MSC (2010): Primary 53A10; Secondary 53C42, 49Q05
DOI: https://doi.org/10.1090/S0002-9947-10-04741-0
Published electronically: March 24, 2010
MathSciNet review: 2608395
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Abstract: In this paper we find, for any arbitrary finite topological type, a compact Riemann surface $ \mathcal{M},$ an open domain $ M\subset\mathcal{M}$ with the fixed topological type, and a conformal complete minimal immersion $ X:M\to\mathbb{R}^3$ which can be extended to a continuous map $ X:\overline{M}\to\mathbb{R}^3,$ such that $ X_{\vert\partial M}$ is an embedding and the Hausdorff dimension of $ X(\partial M)$ is $ 1.$

We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in $ \mathbb{R}^3$, endowed with the topology of the Hausdorff distance.


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Additional Information

Antonio Alarcón
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, Spain
Address at time of publication: Departamento de Matemática Aplicada, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain
Email: ant.alarcon@um.es

DOI: https://doi.org/10.1090/S0002-9947-10-04741-0
Keywords: Complete minimal surfaces, Plateau problem
Received by editor(s): November 16, 2007
Published electronically: March 24, 2010
Additional Notes: The author was partially supported by Spanish MEC-FEDER Grant MTM2007-61775 and Regional J. Andalucía Grant P09-FQM-5088.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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