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Minimal immersions of $ 2$-spheres in $ S\sp{4}$


Author: Ernst A. Ruh
Journal: Proc. Amer. Math. Soc. 28 (1971), 219-222
MSC: Primary 53.75
DOI: https://doi.org/10.1090/S0002-9939-1971-0271880-X
MathSciNet review: 0271880
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Abstract: The classification of isolated singularities of minimal varieties leads to the study of minimal manifolds in the $ n$-sphere. The object of this paper is to show that a minimal $ 2$-sphere in $ {S^4}$ with trivial normal bundle is the standard $ 2$-sphere.


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  • [1] F. J. Almgren, Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math. (2) 84 (1966), 277-292. MR 34 #702. MR 0200816 (34:702)
  • [2] H. Hopf, Über Flächen mit einer Relation zwischen den Hauptkrümmungen, Math. Nachr. 4 (1951), 232-249. MR 12, 634. MR 0040042 (12:634f)
  • [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] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Appl. Math., no. 15, vol. 2, Interscience, New York, 1969. MR 38 #6501. MR 0238225 (38:6501)
  • [5] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105. MR 38 #1617. MR 0233295 (38:1617)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0271880-X
Keywords: Codazzi equations, Euler number, minimal immersion, normal bundle, parallel mean curvature
Article copyright: © Copyright 1971 American Mathematical Society

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