Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Almost spherical convex hypersurfaces

Author: John Douglas Moore
Journal: Trans. Amer. Math. Soc. 180 (1973), 347-358
MSC: Primary 53C45
MathSciNet review: 0320964
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $M$ be a smooth compact hypersurface with positive sectional curvatures in $n$-dimensional euclidean space. This paper gives a sufficient condition for $M$ to lie in the spherical shell bounded by concentric spheres of radius $1 - \epsilon$ and $1 + \epsilon$. This condition is satisfied, in the case where $n = 3$, if the Gaussian curvature or the mean curvature of $M$ is sufficiently pointwise close to one.

References [Enhancements On Off] (What's this?)

  • V. I. Arnol′d and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin, Inc., New York-Amsterdam, 1968. Translated from the French by A. Avez. MR 0232910
  • W. Blaschke, Kreis and Kugel, Viet, Leipzig, 1916; reprint, Chelsea, New York, 1949. MR 17, 887. T. Bonnesen and W. Fenchel, Theorie der Konvexer Körper, Springer, Berlin, 1934; reprint, Chelsea, New York, 1948.
  • Harley Flanders, Differential forms with applications to the physical sciences, Academic Press, New York-London, 1963. MR 0162198
  • Robert Hermann, Differential geometry and the calculus of variations, Mathematics in Science and Engineering, Vol. 49, Academic Press, New York-London, 1968. MR 0233313
  • Dimitri Koutroufiotis, Ovaloids which are almost spheres, Comm. Pure Appl. Math. 24 (1971), 289–300. MR 282318, DOI
  • Heinrich Liebmann, Ueber die Verbiegung der geschlossenen Flächen positiver Krümmung, Math. Ann. 53 (1900), no. 1-2, 81–112 (German). MR 1511083, DOI
  • A. V. Pogorelov, Extrinsic geometry of convex surfaces, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 35. MR 0346714
  • Ju. A. Volkov, Stability of the solution of Minkowski’s problem, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom 18 (1963), no. 1, 33–43 (Russian, with English summary). MR 0146738

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C45

Retrieve articles in all journals with MSC: 53C45

Additional Information

Keywords: Ovaloid, convex hypersurface, almost spherical hypersurface, integral formulae of Minkowski
Article copyright: © Copyright 1973 American Mathematical Society