Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

A note on the asymptotic behaviour of the sum of principal radii of curvature on noncompact complete hypersurfaces


Author: V. I. Oliker
Journal: Proc. Amer. Math. Soc. 77 (1979), 381-384
MSC: Primary 53C40
DOI: https://doi.org/10.1090/S0002-9939-1979-0545600-2
MathSciNet review: 545600
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that under certain hypotheses the following conjecture is correct: on a noncompact complete hypersurface in Euclidean space the two conditions below cannot hold simultaneously: (i) the sum of principal radii of curvature is bounded; (ii) the support function is uniformly continuous.


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

  • [1] S. Cohn-Vossen, Kürzeste Wege and Totalkrümmung auf Flächen, Compositio Math. 2 (1935), 69-133. MR 1556908
  • [2] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1977. MR 0473443 (57:13109)
  • [3] P. Hartman and A. Wintner, On the third fundamental form of a surface, Amer. J. Math. 75 (1953), no. 2, 298-334. MR 0055734 (14:1119c)
  • [4] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and quasi-linear elliptic equations, Academic Press, New York, 1968. MR 0244627 (39:5941)
  • [5] A. L. Verner, The unlimitedness of a hyperbolic horn in the Euclidean space, Sibirsk. Mat. Ž. 11 (1970), 20-29; English transl., Siberian Math. J. 11 (1970), 15-21. MR 0262990 (41:7595)
  • [6] -, On the extrinsic geometry of elementary complete surfaces with nonpositive curvature. I, II, Mat. Sb. 74 (116) (1967), 218-240; 75 (117) (1968), 112-139; English transl., Math. USSR Sb. 3 (1967), 205-224; 4 (1968), 99-123.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C40

Retrieve articles in all journals with MSC: 53C40


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0545600-2
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society