Advertisement

Siberian Mathematical Journal

, 50:757 | Cite as

On the total mean curvature of a nonrigid surface

  • Victor A. Alexandrov
Article

Abstract

Using the Green’s theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field, which immediately yields the following well-known theorem: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.

Keywords

infinitesimal flex vector field flux of a vector field circulation of a vector field Green’s formula 

References

  1. 1.
    Alexandrov A. D., Selected Works. Part II: Intrinsic Geometry of Convex Surfaces, Chapman and Hall/CRC, Boca Raton (2005).Google Scholar
  2. 2.
    Yau S.-T., “Problem section,” in: Seminar on Differential Geometry (Ed. S.-T. Yau); Ann. Math. Stud., 102, pp. 669–706 (1982).Google Scholar
  3. 3.
    Sabitov I. Kh., “A generalized Heron-Tartaglia formula and some of its consequences,” Sb. Math., 189, No. 10, 1533–1561 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Reshetnyak Yu. G., “On nonrigid surfaces of revolution,” Sibirsk Mat. Zh., 3, No. 4, 591–604 (1962).zbMATHGoogle Scholar
  5. 5.
    Sabitov I. Kh., “Local theory on bendings of surfaces,” in: Geometry III. Theory of Surfaces. Encycl. Math. Sci., 48, 1992, pp. 179–250.MathSciNetGoogle Scholar
  6. 6.
    Almgren F. J. and Rivin I., “The mean curvature integral is invariant under bending,” in: The Epstein Birthday Schrift (Eds. I. Rivin et al.), Univ. of Warwick, 1992, pp. 1–21.Google Scholar
  7. 7.
    Schlenker J.-M. and Souam R., “Higher Schläfli formulas and applications,” Compos. Math., 135, No. 1, 1–24 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Souam R., “The Schläfli formula for polyhedra and piecewise smooth hypersurfaces,” Differential Geom. Appl., 20, No. 1, 31–45 (2004).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

Personalised recommendations