  For every $g\in\mathbb{N}_0$ and $\epsilon>0$, we construct a smooth genus
$g$ surface embedded into the unit ball with area $8\pi$ and Willmore energy
smaller than $8\pi + \epsilon$. 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 $\chi_K\leq 0$ 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 $\Omega\Subset\mathbb{R}^3$ with uniformly bounded Willmore energy and
area is compact under varifold convergence.
