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

Minimal surfaces of constant curvature in $ S\sp n$


Author: Robert L. Bryant
Journal: Trans. Amer. Math. Soc. 290 (1985), 259-271
MSC: Primary 53C42; Secondary 53A10
MathSciNet review: 787964
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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}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0787964-8
PII: S 0002-9947(1985)0787964-8
Keywords: Minimal surfaces, Gauss curvature, $ n$-sphere
Article copyright: © Copyright 1985 American Mathematical Society