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

 

Values of Gauss maps of complete minimal surfaces in $ {\mathbb{R}}^{m}$ on annular ends


Authors: Lu Jin and Min Ru
Journal: Trans. Amer. Math. Soc. 359 (2007), 1547-1553
MSC (2000): Primary 53C42; Secondary 32H30
Published electronically: September 12, 2006
MathSciNet review: 2272139
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Abstract: Let $ M$ be a complete minimal surface in $ {\mathbb{R}}^{m}$ and let $ A$ be an annular end of $ M$ which is conformal to $ \{z~\vert~ 0 < 1/r<\vert z\vert<r\}$, where $ z$ is the conformal coordinate. Let $ G$ be the generalized Gauss map of $ M$. We show that $ G(A)$ must intersect every hyperplane in $ {\mathbb{P}}^{m-1}({\mathbb{C}})$, with the possible exception of $ m(m+1)/2$ hyperplanes in general position.


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Additional Information

Lu Jin
Affiliation: Department of Mathematics, Fudan University, Shanghai, People’s Republic of China
Email: jinluk@online.sh.cn

Min Ru
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204
Email: minru@math.uh.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04126-2
PII: S 0002-9947(06)04126-2
Received by editor(s): March 9, 2004
Received by editor(s) in revised form: January 14, 2005
Published electronically: September 12, 2006
Additional Notes: The first author was supported by the National Natural Science Foundation of China (No. 10271029). The second author was supported in part by NSA under grant number H98230-05-1-0042.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.