On complete graphs with negative r-mean curvature

Elbert, Maria

doi:10.1090/S0002-9939-00-05671-9

On complete graphs with negative r-mean curvature
HTML articles powered by AMS MathViewer

by Maria Fernanda Elbert PDF

Proc. Amer. Math. Soc. 128 (2000), 1443-1450 Request permission

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

Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Shiing-shen Chern, On the curvatures of a piece of hypersurface in euclidean space, Abh. Math. Sem. Univ. Hamburg 29 (1965), 77–91. MR 188949, DOI 10.1007/BF02996311
S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
Takashi Okayasu, $\textrm {O}(2)\times \textrm {O}(2)$-invariant hypersurfaces with constant negative scalar curvature in $E^4$, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1045–1050. MR 990430, DOI 10.1090/S0002-9939-1989-0990430-7
Robert C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J. 20 (1973), 373–383. MR 334045
Harold Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211–239. MR 1216008
Isabel Maria da Costa Salavessa, Graphs with parallel mean curvature, Proc. Amer. Math. Soc. 107 (1989), no. 2, 449–458. MR 965247, DOI 10.1090/S0002-9939-1989-0965247-X
Brian Smyth and Frederico Xavier, Efimov’s theorem in dimension greater than two, Invent. Math. 90 (1987), no. 3, 443–450. MR 914845, DOI 10.1007/BF01389174