Confined structures of least bending energy

TL;DR

This paper investigates the minimal Willmore energy of smooth sphere embeddings within the unit ball under surface area constraints, revealing nonconvex structures and sharp energy increases when surface area slightly exceeds that of a sphere.

Contribution

It provides new bounds and insights into the shape and energy behavior of constrained minimal surfaces, especially near the sphere's surface area.

Findings

Minimal Willmore energy depends on surface area with sharp increases near the sphere's area.

Minimizing surfaces become nonconvex when surface area slightly exceeds that of the sphere.

Energy increases at a square root rate with respect to surface area increase.

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.