Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-03-07332-5
Published electronically: December 19, 2003
MathSciNet review: 2053332
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. J. L. Barbosa and M. do Carmo, On the size of a stable minimal surface in $\textbf{R}^3$. Amer. J. Math. 98 (1976), 515-528. MR 54:1292
  • 2. M. do Carmo and C. K. Peng, Stable complete minimal surfaces in $\textbf{R}^3$ are planes. Bull. Amer. Math. Soc. (N.S.) 1 (1979), 903-906. MR 80j:53012
  • 3. D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 33 (1980), 199-211. MR 81i:53044
  • 4. H. B. Lawson, Jr., Lectures on minimal submanifolds, Vol. I, second edition, Publish or Perish, Wilmington, DE, 1980. MR 82d:53035b
  • 5. D. A. Hoffman and R. Osserman, The geometry of the generalized Gauss map. Mem. Amer. Math. Soc. vol. 28, no. 236, 1980. MR 82b:53012
  • 6. M. Micallef, Stable minimal surfaces in Euclidean space. J. Differential Geometry 19 (1984), 57-84. MR 85e:53009
  • 7. W. Wirtinger, Eine determinanteindentit$\ddot{a}$t und ihre anwendung auf analytische gebilde und Hermitesche massbestimmung. Monatsh. Math. Physik 44 (1936), 343-365.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 49Q05, 53A10

Retrieve articles in all journals with MSC (2000): 49Q05, 53A10


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

American Mathematical Society