Confined structures of least bending energy

Abstract

In this paper we study a constrained minimization problem for the Willmore functional. For prescribed surface area, we consider smooth embeddings of the sphere into the unit ball. We evaluate the dependence of the the minimal Willmore energy of such surfaces on the prescribed surface area and prove corresponding upper and lower bounds. Interesting features arise when the prescribed surface area just exceeds the surface area of the unit sphere. We show that (almost) minimizing surfaces cannot be a $C^2$-small perturbation of the sphere. Indeed, they have to be nonconvex and there is a sharp increase in Willmore energy with a square root rate with respect to the increase in surface area.