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manuscripta mathematica

, Volume 59, Issue 3, pp 277–294 | Cite as

New examples of imbedded spherical soap bubbles inS N (1)

  • Ivan Sterling
Article
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Abstract

A. D. Alexandrov proved that the only imbedded soap bubbles in euclidean or hyperbolicn-space are round spheres. This also holds for ann-dimensional hemisphere. In this paper we give many new examples of differentiable (n−1)-spheres imbedded as soap bubbles in sphericaln-space. Thus we show Alexandrov's theorem is false for sphericaln-space even under the added assumption that the soap bubbles are differentiable (n−1)-spheres.

Keywords

Number Theory Algebraic Geometry Topological Group Round Sphere Soap Bubble 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    ALEXANDROV, A. D.: Uniqueness theorems for surfaces in the largeV. Vestnik Leningrad Univ.13, No. 19, 5–8 (1958)Google Scholar
  2. [2]
    ALEXANDROV, A. D.: A characteristic property of spheres. Annali di Mat. Pura Ed. Appl.58, 303–315 (1962)Google Scholar
  3. [3]
    BACK, A., DoCARMO, M., HSIANG, W. Y.: On fundamental equations of equivariant differential geometry (to appear)Google Scholar
  4. [4]
    CHERN, S. S.: Differential geometry, its past and its future. Actes Cong. Intern. Math. Tome1, 41–53 (1970)Google Scholar
  5. [5]
    CHERN, S. S.: On surfaces of constant mean curvature in a three-dimensional space of constant curvature. Geometric Dynamics, Lecture Notes in Math., vol. 1007, 104–108: Springer-Verlag 1983Google Scholar
  6. [6]
    DELAUNAY, C.: Sur la surface de revolution dont la courbure mayenne est constante. J. Math. Pures Appl. Ser. 1,6, 309–320 (1841)Google Scholar
  7. [7]
    HSIANG, W. T., HSIANG, W. Y., STERLING, I.: On the construction of codimension two minimal immersions of exotic spheres into euclidean spheres. Invent. Math.82, 447–460 (1985)Google Scholar
  8. [8]
    HSIANG, W. Y.: Minimal cones and the spherical Bernstein problem, I. Ann. of Math.118, 63–71 (1983)Google Scholar
  9. [9]
    HSIANG, W. Y.: Minimal cones and the spherical Bernstein problem, II. Invent. Math.74, 351–369 (1983)Google Scholar
  10. [10]
    HSIANG, W. Y.: OnW-hypersurfaces of generalizeed rotational types inE n+1, I. (to appear)Google Scholar
  11. [11]
    HSIANG, W. Y.: On generalization of theorems of A. D. Alexandrov and C. Delaunay on hypersurfaces of constant mean curvature. Duke J. of Math.49, no. 3, 485–496 (1982)Google Scholar
  12. [12]
    HSIANG, W. Y.: On rotationalW-hypersurfaces in spaces of constant curvature and generalized laws of sine and cosine. Bull. of the Inst. of Math. Acad. Sinica.11, no. 3, 349–373 (Sept. 1983)Google Scholar
  13. [13]
    HSIANG, W. Y.: Generalized rotational hypersurfaces of constant mean curvature in the euclidean spaces, I. Jour. of Differential Geom.17, 337–356 (1982)Google Scholar
  14. [14]
    HSIANG, W. Y., LAWSON, B. H., Jr.: Minimal submanifolds of low cohomogeneity. Jour. of Differential Geom.5, 1–38 (1971)Google Scholar
  15. [15]
    HSIANG, W. Y., STERLING, I.: On the construction of non-equatorial minimal hyperspheres inS n (1) with stable cones inR n +1. Proc. National Acad. Sci. (USA)81, 8035–8036 (Dec. 1984)Google Scholar
  16. [16]
    HSIANG, W. Y., STERLING, I.: Minimal cones and the spherical Bernstein problem, III. Invent. Math.85, 223–247 (1986)Google Scholar
  17. [17]
    HSIANG, W. Y., TOMTER, P.: On the existence of infinitely many mutually non-congruent minimal immersions ofS n +1 intoS n (1),n≥4. (to appear)Google Scholar
  18. [18]
    HSIANG, W. Y., YU, W.: A generalization of a theorem of Delaunay. Jour. of Differential Geom.16, 161–177 (1981)Google Scholar
  19. [19]
    KARCHER, H., FERUS, D.: Non-rotational minimal spheres and minimizing cones. Comment. Math. Helv.60, no. 2, 247–269 (1985)Google Scholar
  20. [20]
    STERLING, I.: A generalization of a theorem of Delaunay to rotationalW-hypersurfaces of σltype inH n+1 andS n+1. Pacific J. Math.126, no. 2 (1987)Google Scholar
  21. [21]
    TOMTER, P.: The spherical Bernstein problem in even dimensions. Bull. Amer. Math. Soc. (N.S.)11, 183–185 (1984)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Ivan Sterling
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaSanta CruzUSA
  2. 2.Fachbereich 3-MathematikTU BerlinBerlin 12

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