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Siberian Mathematical Journal

, Volume 45, Issue 1, pp 19–52 | Cite as

Irregular C1,β-Surfaces with an Analytic Metric

  • Yu. F. Borisov
Article

Abstract

We prove that in the class C1,β with β<1/13 it is possible to continuously deform an analytic convex surface of positive Gaussian curvature (or a plane) so as to lose boundedness of the extrinsic curvature in the Pogorelov sense. We demonstrate how to replace the bound β<1/13 with β<1/7.

continuous deformation analytic surface positive Gaussian curvature local convexity 

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References

  1. 1.
    Alexandrov A. D., Intrinsic Geometry of Convex Surfaces [in Russian], GITTL, Moscow; Leningrad (1948).Google Scholar
  2. 2.
    Alexandrov A. D. and Zalgaller V. A., “Two-dimensional manifolds of bounded curvature. I,” Trudy Mat. Inst. Steklov, 63, 3-259 (1962).Google Scholar
  3. 3.
    Alexandrov A. D. and Zalgaller V. A., “Two-dimensional manifolds of bounded curvature. II,” Trudy Mat. Inst. Steklov, 76, 1-152 (1965).Google Scholar
  4. 4.
    Pogorelov A. V., Extrinsic Geometry of Convex Surfaces, Nauka, Moscow (1989).Google Scholar
  5. 5.
    Borisov Yu. F. and Shefel' S. Z., “Surfaces with bounded extrinsic curvature and positive Gauss curvature,” Dokl. Akad. Nauk SSSR, 200,No. 2, 259-261 (1971).Google Scholar
  6. 6.
    Nash J., “C′-isometric embeddings,” Matematika, 1,No. 2, 3-16 (1957).Google Scholar
  7. 7.
    Kuiper N. H., “On C′-isometric embeddings,” Matematika, 1,No. 2, 17-28 (1957).Google Scholar
  8. 8.
    Borisov Yu. F., “Parallel displacement on a smooth surface. I,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 2,No. 7, 160-171 (1958).Google Scholar
  9. 9.
    Borisov Yu. F., “Parallel displacement on a smooth surface. II,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 4,No. 19, 45-54 (1958).Google Scholar
  10. 10.
    Borisov Yu. F., “Parallel displacement on a smooth surface. III,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 1,No. 1, 34-50 (1959).Google Scholar
  11. 11.
    Borisov Yu. F., “On the problem of parallel displacement on a smooth surface and connection between the extrinsic shape of a smooth surface and intrinsic geometry,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 4,No. 19, 127-129 (1960).Google Scholar
  12. 12.
    Borisov Yu. F., “On connection between the extrinsic shape of a smooth surface and intrinsic geometry,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 2,No. 13, 20-25 (1959).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Yu. F. Borisov
    • 1
  1. 1.Sobolev Institute of MathematicsNovosibirsk

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