Minimal surfaces of constant curvature in 𝑆ⁿ

Bryant, Robert L.

doi:10.1090/S0002-9947-1985-0787964-8

Minimal surfaces of constant curvature in $S^ n$
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by Robert L. Bryant

Trans. Amer. Math. Soc. 290 (1985), 259-271

DOI: https://doi.org/10.1090/S0002-9947-1985-0787964-8

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Abstract:

In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in ${S^n}$ of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [Calabi], while the solution to the local problem was found by [Wallach]. The case of nonpositive Gaussian curvature is more subtle and has remained open. We prove that there are no minimal surfaces in ${S^n}$ of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in ${S^n}$ and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into ${S^n}$.

References

Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111–125. MR 233294
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
Katsuei Kenmotsu, Minimal surfaces with constant curvature in $4$-dimensional space forms, Proc. Amer. Math. Soc. 89 (1983), no. 1, 133–138. MR 706526, DOI 10.1090/S0002-9939-1983-0706526-5

Lectures on minimal submanifolds

M. Pinl, Minimalflächen fester Gaussscher Krümmung, Math. Ann. 136 (1958), 34–40 (German). MR 97825, DOI 10.1007/BF01350284
James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
Nolan R. Wallach, Extension of locally defined minimal immersions into spheres, Arch. Math. (Basel) 21 (1970), 210–213. MR 271878, DOI 10.1007/BF01220905