On the Willmore energy of shells under infinitesimal deformations

Abstract

In this paper we apply tensor calculus and differential geometry to consider shell structure. Using tensor calculus we examine the stationarity of the Willmore energy under infinitesimal deformations of the surfaces in ℜ 3 . We obtain the class of the surfaces which does not change its Willmore energy under infinitesimal deformations. In particular, a special kind of deformation is considered—infinitesimal bending which preserves the arc length. The change of the Willmore energy under such deformations is determined. Also, we give a new proof of a well-known theorem (that reads that the total mean curvature of a surface is stationary under an infinitesimal bending), applying tensor calculus.