Minimal surfaces with constant curvature in $4$-dimensional space forms
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- by Katsuei Kenmotsu PDF
- Proc. Amer. Math. Soc. 89 (1983), 133-138 Request permission
Abstract:
We classify minimal surfaces with constant Gaussian curvature in a $4$-dimensional space form without any global assumption. As a corollary of the main theorem, we show there is no isometric minimal immersion of a surface with constant negative Gaussian curvature into the unit $4$-sphere even locally. This gives a partial answer to a problem proposed by S. T. Yau.References
- João Lucas Marquês Barbosa, On minimal immersions of $S^{2}$ into $S^{2m}$, Trans. Amer. Math. Soc. 210 (1975), 75–106. MR 375166, DOI 10.1090/S0002-9947-1975-0375166-2
- Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111–125. MR 233294
- Bang-yen Chen, Minimal surfaces with constant Gauss curvature, Proc. Amer. Math. Soc. 34 (1972), 504–508. MR 296828, DOI 10.1090/S0002-9939-1972-0296828-4
- Shiing Shen Chern, On the minimal immersions of the two-sphere in a space of constant curvature, Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 27–40. MR 0362151
- Manfredo P. do Carmo and Nolan R. Wallach, Minimal immersions of spheres into spheres, Ann. of Math. (2) 93 (1971), 43–62. MR 278318, DOI 10.2307/1970752
- Luthur Pfahler Eisenhart, An Introduction to Differential Geometry, Princeton Mathematical Series, vol. 3, Princeton University Press, Princeton, N. J., 1940. MR 0003048
- Katsuei Kenmotsu, On compact minimal surfaces with non-negative Gaussian curvature in a space of constant curvature. II, Tohoku Math. J. (2) 27 (1975), no. 3, 291–301. MR 514824, DOI 10.2748/tmj/1203529241
- Katsuei Kenmotsu, On minimal immersions of $R^{2}$ into $S^{N}$, J. Math. Soc. Japan 28 (1976), no. 1, 182–191. MR 405218, DOI 10.2969/jmsj/02810182
- Tominosuke Ôtsuki, Minimal submanifolds with $m$-index $2$ and generalized Veronese surfaces, J. Math. Soc. Japan 24 (1972), 89–122. MR 295259, DOI 10.2969/jmsj/02410089
- Nolan R. Wallach, Extension of locally defined minimal immersions into spheres, Arch. Math. (Basel) 21 (1970), 210–213. MR 271878, DOI 10.1007/BF01220905
- Yung-Chow Wong, Contributions to the theory of surfaces in a 4-space of constant curvature, Trans. Amer. Math. Soc. 59 (1946), 467–507. MR 16231, DOI 10.1090/S0002-9947-1946-0016231-0
- Shing Tung Yau (ed.), Seminar on Differential Geometry, Annals of Mathematics Studies, No. 102, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979–1980. MR 645728
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 89 (1983), 133-138
- MSC: Primary 53C42
- DOI: https://doi.org/10.1090/S0002-9939-1983-0706526-5
- MathSciNet review: 706526