Minimizers of Generalized Willmore Functionals

TL;DR

This paper introduces generalized Willmore functionals inspired by physics and biology, proves existence of minimizers in complex surface classes, and extends regularity results to these generalized settings.

Contribution

It develops a framework for generalized Willmore functionals, establishes existence of minimizers in stratified surface classes, and extends regularity theory to these new functionals.

Findings

Existence of area constrained minimizers in bubble tree classes.

Existence of minimal membranes with bending energies.

Regularity results extend to generalized Willmore surfaces.

Abstract

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.