On the Canham Problem: Bending Energy Minimizers for any Genus and Isoperimetric Ratio

TL;DR

This paper proves the existence of genus-$g$ surfaces in $ ^3$ that minimize bending energy for any given isoperimetric ratio, extending previous work and constructing explicit comparison surfaces.

Contribution

It demonstrates the existence of bending energy minimizers for all genus and isoperimetric ratios by constructing specific comparison surfaces using catenoidal bridges.

Findings

Existence of energy minimizers for all genus and isoperimetric ratios.

Construction of comparison surfaces with arbitrarily small isoperimetric ratio.

Bending energy of constructed surfaces is less than 8π.

Abstract

Building on work of Mondino-Scharrer, we show that among closed, smoothly embedded surfaces in $\mathbb{R}^3$ of genus $g$ and given isoperimetric ratio $v$, there exists one with minimum bending energy $\mathcal{W}$. We do this by gluing $g+1$ small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation $\Delta (\Delta +2)\varphi=0$ on the $(g+1)$-punctured sphere $\mathbb{S}^2$ to construct a comparison surface of genus $g$ with arbitrarily small isoperimetric ratio $v\in (0, 1)$ and $\mathcal{W} < 8\pi$.