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 $${\mathbb {R}}^3$$ 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 a singular solution for linearized Willmore equation $$\Delta (\Delta +2)\varphi =0$$ $$(g+1)$$ -punctured sphere {S}}^2$$ construct comparison surface arbitrarily $$v\in (0, 1)$$ {W}}< 8\pi $$