  We consider the problem of minimizing the Willmore energy connected surfaces
with prescribed surface area which are confined to a finite container. To this
end, we approximate the surface by a phase field function $u$ taking values
close to +1 on the inside of the surface and -1 on its outside. The confinement
of the surface is now simply given by the domain of definition of $u$. A
diffuse interface approximation for the area functional, as well as for the
Willmore energy are well known. We address the topological constraint of
connectedness by a nested minimization of two phase fields, the second one
being used to identify connected components of the surface. In this article, we
provide a proof of Gamma-convergence of our model to the sharp interface limit.
