On an inequality of T. J. Willmore

Abstract

Willmore proved that the integral of the square of mean curvature H H over a closed surface M 2 {M^2} in E 3 , ∫ M 2 H 2 d V {E^3},{\smallint _{{M^2}}}{H^2}dV , is ≧ 4 π \geqq 4\pi , and equal to 4 π 4\pi when and only when M 2 {M^2} is a sphere in E 3 {E^3} . In this paper we give some generalizations of Willmore’s result.