Change of the Willmore energy under infinitesimal bending of membranes

Abstract

Change of the Willmore energy, as the special case of elastic bending energy, under infinitesimal bending of the vesicle membrane is discussed. A membrane is thought of as a smooth surface in R 3 because its thickness is much smaller than its lateral dimension. Variation of the Willmore energy at the surface point under infinitesimal bending of that surface, as well as the condition for the stationarity of the Willmore energy are given. Some examples are visualized. Variation of the Willmore energy of a compact path-connected smooth surface in the Euclidean 3-space is reduced to a line integral of a special vector field and it is found that the Willmore energy of a boundary-free surface is stationary under an infinitesimal bending. Also, the Willmore energy of a minimal surface is stationary under its infinitesimal bending.