Confining spheres within hyperspheres

TL;DR

This paper develops a framework to analyze the equilibrium shapes of deformable surfaces confined within smaller hyperspheres, revealing that small-area surfaces deform harmonically with specific stable and unstable modes.

Contribution

It introduces a new theoretical framework for understanding the equilibrium configurations of confined deformable surfaces in higher dimensions.

Findings

Ground state is a quadrupole deformation of a sphere

Higher multipole deformations are unstable

Harmonic deformations describe equilibrium states for small excess area

Abstract

The bending energy of any freely deformable closed surface is quadratic in its curvature. In the absence of constraints, it will be minimized when the surface adopts the form of a round sphere. If the surface is confined within a hypersurface of smaller size, however, this spherical state becomes inaccessible. A framework is introduced to describe the equilibrium states of the confined surface. It is applied to a two-dimensional surface confined within a three-dimensional hypersphere of smaller radius. If the excess surface area is small, the equilibrium states are represented by harmonic deformations of a two-sphere: the ground state is described by a quadrupole; all higher multipoles are shown to be unstable.