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

Author(s): Francisco Martin; Masaaki Umehara; Kotaro Yamada
Journal: Proc. Amer. Math. Soc. 137 (2009), 3437-3450.
MSC (2000): Primary 53A10, 32H02; Secondary 53C42, 53C50
Posted: June 1, 2009
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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: 10.1090/S0002-9939-09-09953-5
PII: S 0002-9939(09)09953-5
Received by editor(s): October 26, 2008
Posted: June 1, 2009
Communicated by: Richard A. Wentworth
Copyright of article: Copyright 2009, American Mathematical Society

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