Complete bounded holomorphic curves immersed in 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

DOI:
https://doi.org/10.1090/S0002-9939-09-09953-5

Published electronically:
June 1, 2009

MathSciNet review:
2515413

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Abstract | References | Similar Articles | Additional Information

Abstract: Recently, a complete holomorphic immersion of the unit disk into 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 .

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:
https://doi.org/10.1090/S0002-9939-09-09953-5

Received by editor(s):
October 26, 2008

Published electronically:
June 1, 2009

Communicated by:
Richard A. Wentworth

Article copyright:
© Copyright 2009
American Mathematical Society