  We minimise the Canham-Helfrich energy in the class of closed immersions with
prescribed genus, surface area and enclosed volume. Compactness is achieved in
the class of oriented varifolds. The main result is a lower-semicontinuity
estimate for the minimising sequence, which is in general false by a counter
example by Gro{\ss}e-Brauckmann. The main argument involved is showing partial
regularity of the limit. It entails comparing the Helfrich energy of the
minimising sequence locally to that of a biharmonic graph. This idea is by
Simon, but it cannot be directly applied, since the area and enclosed volume of
the graph may differ. By an idea of Schygulla we adjust these quantities by
using a two parameter diffeomorphism of $\mathbb{R}^3$
