Acta mathematica scientia, Series B >
ON THE WILLMORE'S THEOREM FOR CONVEX HYPERSURFACES
Received date: 2007-07-05
Online published: 2011-03-20
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Supported in part by CNSF (10671197)
Let M be a compact convex hypersurface of class C2, which is assumed to bound a nonempty convex body K in the Euclidean space Rn and H be the mean curvature of M. We obtain a lower bound of the total square of mean curvature ∫M H2dA. The bound is the Minkowski quermassintegral of the convex body K. The total square of mean curvature attains the lower bound when M is an (n-1)-sphere.
ZHOU Jia-Zu . ON THE WILLMORE'S THEOREM FOR CONVEX HYPERSURFACES[J]. Acta mathematica scientia, Series B, 2011 , 31(2) : 361 -366 . DOI: 10.1016/S0252-9602(11)60237-3
[1] Burago Yu D, Zalgaller V A. Geometric Inequalities. Springer-Verlag Berlin Heidelberg, 1998
[2] Chen B Y. Geometry of Submanifolds. New York: Marcel Dekker Inc, 1973
[3] Chen B Y. Total Mean Curvature and Submanifolds of Finite Type. Singapore: World Scientific, 1984
[4] Li S, Yau S T. A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent Math, 1982, 69: 269--291
[5] Ren D. Topics in Integral Geometry. Sigapore: World Scientific, 1994
[6] SantalòL A. Integral Geometry and Geometric Probability. Addison-Wesley, Reading, Mass, 1976
[7] Schneider R. Convex bodies: the Brunn-Minkowski theory. Cambridge: Cambridge Univ Press, 1993
[8] Willmore T. Total Curvature in Riemannian Geometry. Chichester: Ellis Horwood, 1982
[9] Zhou J. On the Willmore deficit of convex surfaces. Lectures in Applied Mathematics of the Amer Math Soc, 1994, 30: 279--287
[10] Zhou J. The Willmore functional and the containment problem in R4. Science in China Series A: Mathematics, 2007, 50(3): 325--333
[11] Zhou J. On Willmore functional for submanifolds. Canad Math Bull, 2007, 50(3): 474--480
[12] Zhou J, Jiang D, Li M, Chen F. On Ros' Theorem for Hypersurface. Acta Math Sinica, 2009, 52(6): 1075--1084
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