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

 

Stable minimal surfaces in $\textbf{R}^4$ with degenerate Gauss map


Author: Toshihiro Shoda
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1285-1293
MSC (2000): Primary 49Q05, 53A10
Published electronically: December 19, 2003
MathSciNet review: 2053332
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Abstract: A complete oriented stable minimal surface in $\textbf{R}^3$ is a plane, but in $\textbf{R}^4$, there are many non-flat examples such as holomorphic curves. The Gauss map plays an important role in the theory of minimal surfaces. In this paper, we prove that a complete oriented stable minimal surface in $\textbf{R}^4$ with $\alpha$-degenerate Gauss map (for $\alpha > 1/4$) is a plane.


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Additional Information

Toshihiro Shoda
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo, 152-8551, Japan
Email: tshoda@math.titech.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07332-5
PII: S 0002-9939(03)07332-5
Received by editor(s): March 6, 2000
Published electronically: December 19, 2003
Communicated by: Bennett Chow
Article copyright: © Copyright 2003 American Mathematical Society