Acta mathematica scientia,Series B ›› 2011, Vol. 31 ›› Issue (2): 361-366.doi: 10.1016/S0252-9602(11)60237-3

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ON THE WILLMORE'S THEOREM FOR CONVEX HYPERSURFACES

 ZHOU Jia-Zu   

  1. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Southeast Guizhou Vocational College of Technology for Nationalities, Kaili |556000, China
  • Received:2007-07-05 Online:2011-03-20 Published:2011-03-20
  • Supported by:

    Supported in part by CNSF (10671197)

Abstract:

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.

Key words: Mean curvature,  the Willmore deficit, Minkowski quermassintegrale

CLC Number: 

  • 52A20| 
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