  We introduce a notion of generalized Willmore functionals motivated by the
Hawking energy of General Relativity and bending energies of membranes. An
example of a bending energy is discussed in detail. Using results of Y. Chen
and J. Li, we present a compactness result for branched, immersed, haunted,
stratified surface with bounded area and Willmore energy. This allows us to
prove the existence of area constrained minimizers for generalized Willmore
functionals in the class of haunted, branched, immersed bubble trees by direct
minimization. Here a haunted, stratified surfaces are introduced, in order to
account for bubbling and vanishing components along the minimization process.
Similarly, we obtain the existence of area and volume constrained, minimal,
closed membranes for the discussed bending energy. Moreover, we argue that the
regularity results of A. Mondino and T. Rivi\`ere for Willmore surfaces can be
carried over to the setting of generalized Willmore surfaces. In particular,
this means that critical points of a generalized Willmore functional are smooth
away from finitely many points.
