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

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 $\mathrm{\Delta }(\mathrm{\Delta }+2)\phi =0$ 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 $\mathcal{W}<8\pi$.