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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Complete bounded holomorphic curves immersed in $ \mathbb{C}^2$ with arbitrary genus


Authors: Francisco Martin, Masaaki Umehara and Kotaro Yamada
Journal: Proc. Amer. Math. Soc. 137 (2009), 3437-3450
MSC (2000): Primary 53A10, 32H02; Secondary 53C42, 53C50
Published electronically: June 1, 2009
MathSciNet review: 2515413
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Abstract: Recently, a complete holomorphic immersion of the unit disk $ \mathbb{D}$ into $ \mathbb{C}^2$ whose image is bounded was constructed by the authors. In this paper, we shall prove the existence of complete holomorphic null immersions of Riemann surfaces with arbitrary genus and finite topology whose image is bounded in $ \mathbb{C}^2$.

As an analogue to the above construction, we also give a new method to construct complete bounded minimal immersions (resp. weakly complete maximal surfaces) with arbitrary genus and finite topology in Euclidean 3-space (resp. Lorentz-Minkowski 3-spacetime).


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Additional Information

Francisco Martin
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Email: fmartin@ugr.es

Masaaki Umehara
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: umehara@math.sci.osaka-u.ac.jp

Kotaro Yamada
Affiliation: Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japan
Email: kotaro@math.kyushu-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09953-5
PII: S 0002-9939(09)09953-5
Received by editor(s): October 26, 2008
Published electronically: June 1, 2009
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2009 American Mathematical Society