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)

 
 

 

On complete graphs with negative r-mean curvature


Author: Maria Fernanda Elbert
Journal: Proc. Amer. Math. Soc. 128 (2000), 1443-1450
MSC (2000): Primary 53C42; Secondary 53A10
DOI: https://doi.org/10.1090/S0002-9939-00-05671-9
Published electronically: February 7, 2000
MathSciNet review: 1712913
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We generalize Efimov's Theorem for graphs in Euclidean space using the scalar curvature, with an additional hypothesis on the second fundamental form.


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

  • [BC] J.L.Barbosa and A.G.Colares, Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15, 277-297 (1997). MR 98h:52091
  • [Cha] I. Chavel, Eigenvalues in Riemmanian Geometry. Academic Press, Orlando, USA (1984). MR 86g:58140
  • [Che] S.S.Chern, On the curvatures of a piece of hypersurface in Euclidean space. Abh. Math. Sem. Univ. Hamburg 29, 77-91 (1965). MR 32:6376
  • [CY] S. Y. Cheng and S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math 28, 333-354 (1975). MR 52:6608
  • [O] T.Okayasu, $O(2)\times O(2)$-Invariant hypersurfaces with constant negative scalar curvature in $E^{4}$. Proc. Amer. Math. Soc. 107, 4, 1045-1050 (1989). MR 90f:53094
  • [Re] R.C.Reilly, On the Hessian of a function and the curvatures of its graph. Michigan Math. J. 20, 373-383 (1973). MR 48:12364
  • [Ro] H.Rosenberg, Hypersurfaces of constant curvatures in space forms. Bull. Sc. Math. 117, 211-239 (1993). MR 94b:53097
  • [Sa] I.M.C.Salavessa, Graphs with parallel mean curvature. Proc. Amer. Math. Soc. 107, 2, 449-458 (1989). MR 90a:53072
  • [SX] B.Smyth and F.Xavier, Efimov's Theorem in dimension greater than two. Invent. Math. 90, 443-450 (1987). MR 89h:53014

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C42, 53A10

Retrieve articles in all journals with MSC (2000): 53C42, 53A10


Additional Information

Maria Fernanda Elbert
Affiliation: Instituto de Matematica, UFRJ, Cx. Postal 68530, 21941-590 Rio de Janeiro, RJ, Brasil
Address at time of publication: IMPA - Estrada Dona Castorina, 110, 22460-320 - Rio de Janeiro, RJ, Brasil
Email: elbert@impa.br

DOI: https://doi.org/10.1090/S0002-9939-00-05671-9
Keywords: Negative r-mean curvature, complete graphs, divergence, Cheeger constant
Received by editor(s): June 17, 1998
Published electronically: February 7, 2000
Communicated by: Christopher Croke
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society