Helfrich's Energy and Constrained Minimisation

For every $g\in {\mathbb{N}}_0$ and $ϵ>0$, we construct a smooth genus $g$ surface embedded into the unit ball with area $8\pi$ and Willmore energy smaller than $8\pi +ϵ$. From this we deduce that a minimising sequence for Willmore’s energy in the class of genus $g$ surfaces embedded in the unit ball with area $8\pi$ converges to a doubly covered sphere for all $g\in {\mathbb{N}}_0$. We obtain the same result for certain Canham-Helfrich energies with ${\chi }_K\le 0$ without genus constraint and show that Canham-Helfrich energies with ${\chi }_K>0$ are not bounded from below in the class of smooth surfaces with area $S$ embedded into a domain $\Omega\Subset $\mathbb{R}$^3$. Furthermore, we prove that the class of connected surfaces embedded in a domain $\mathrm{\Omega }⋐{\mathbb{R}}^3$ with uniformly bounded Willmore energy and area is compact under varifold convergence.