  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
$\mathcal{W} < 8\pi$.
