Embedded surfaces of arbitrary genus minimizing the Willmore energy under isoperimetric constraint

TL;DR

This paper investigates the problem of finding embedded surfaces of any genus in three-dimensional space that minimize the Willmore energy while maintaining a fixed isoperimetric ratio, extending previous work done for spherical surfaces.

Contribution

It generalizes the minimization of Willmore energy under isoperimetric constraints to surfaces of arbitrary genus, building on prior solutions for genus zero surfaces.

Findings

Addresses minimization for surfaces of any genus

Extends previous genus zero results to higher genus

Provides new methods for constrained surface optimization

Abstract

The isoperimetric ratio of an embedded surface in $R^3$ is defined as the ratio of the area of the surface to power three to the squared enclosed volume. The aim of the present work is to study the minimization of the Willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus $g\geq 0$. The corresponding problem in the case of spherical surfaces, i.e. $g=0$, was recently solved by Schygulla with different methods.