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Proceedings of the American Mathematical Society

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

 

 

Minimal disks and compact hypersurfaces in Euclidean space


Authors: John Douglas Moore and Thomas Schulte
Journal: Proc. Amer. Math. Soc. 94 (1985), 321-328
MSC: Primary 53C40; Secondary 53C42
DOI: https://doi.org/10.1090/S0002-9939-1985-0784186-7
MathSciNet review: 784186
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Abstract: Let $ {M^n}$ be a smooth connected compact hypersurface in $ (n + 1)$-dimensional Euclidean space $ {E^{n + 1}}$, let $ {A^{n + 1}}$ be the unbounded component of $ {E^{n + 1}} - {M^n}$, and let $ {\kappa _1} \leqslant {\kappa _2} \leqslant \cdots \leqslant {\kappa _n}$ be the principal curvatures of $ {M^n}$ with respect to the unit normal pointing into $ {A^{n + 1}}$. It is proven that if $ {\kappa _2} + \cdots + {\kappa _n} < 0$, then $ {A^{n + 1}}$ is simply connected.


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  • [C] R. Courant, The existence of a minimal surface of least area bounded by prescribed Jordan arcs and prescribed surfaces, Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 97-101.
  • [CD] R. Courant and N. Davids, Minimal surfaces spanning closed manifolds, Proc. Nat. Acad. Sci. U. S. A. 26 (1940), 194–199. MR 0001472
  • [HW] R. Howard and H. Wei, On the existence and nonexistence of stable submanifolds and currents in positively curved manifolds and the topology of submanifolds in Euclidean spaces (to appear).
  • [J] Willi Jäger, Behavior of minimal surfaces with free boundaries, Comm. Pure Appl. Math. 23 (1970), 803–818. MR 0266067, https://doi.org/10.1002/cpa.3160230508
  • [KN] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1393940
    Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1969 original; A Wiley-Interscience Publication. MR 1393941
  • [L] H. B. Lawson, Lectures on minimal submanifolds, Vol. 1, Publish or Perish, Berkeley, Calif., 1980.
  • [MY] William H. Meeks III and Shing Tung Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980), no. 3, 441–484. MR 595203, https://doi.org/10.2307/1971088

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0784186-7
Article copyright: © Copyright 1985 American Mathematical Society