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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Minimal hypersurfaces in an $ m$-sphere

Author: Bang-yen Chen
Journal: Proc. Amer. Math. Soc. 29 (1971), 375-380
MSC: Primary 53.04
MathSciNet review: 0285999
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Abstract: (1) A submanifold $ {M^n}$ of a euclidean space $ {E^{n + 2}}$ of codimension 2 is a pseudo-umbilical submanifold with constant mean curvature if and only if it is a minimal hypersurface of a hypersphere of $ {E^{n + 2}}$. (2) A complete oriented minimal surface $ {M^2}$ of a 3-sphere $ {S^3}$ on which the Gauss curvature does not change its sign is either an equatorial sphere or a Clifford flat torus.

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

  • [1] B.-y. Chen, On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof (to appear). MR 0291994 (45:1082)
  • [2] T. Itoh, Complete surfaces in $ {E^4}$ with constant mean curvature, Kōdai Math. Sem. Rep. 22 (1970), 150-158. MR 0267496 (42:2398)
  • [3] H. B. Lawson, Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187-197. MR 38 #6505. MR 0238229 (38:6505)
  • [4] R. Osserman, Minimal varieties, Bull. Amer. Math. Soc. 75 (1969), 1092-1120. MR 0276875 (43:2615)

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Keywords: Second fundamental form, mean curvature vector, mean curvature, pseudo-umbilical submanifold, minimal hypersurface, equatorial sphere, Clifford flat torus, Gauss curvature
Article copyright: © Copyright 1971 American Mathematical Society

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