Energy quantization for constrained Willmore surfaces

TL;DR

This paper proves energy quantization and strong compactness results for constrained Willmore surfaces under certain geometric constraints and energy bounds.

Contribution

It establishes energy quantization and compactness theorems for constrained Willmore surfaces with area, volume, and mean curvature constraints.

Findings

Energy quantization for constrained Willmore surfaces.

Strong compactness under energy thresholds.

Compactness of minimizers in specific problems.

Abstract

We establish an energy quantization for constrained Willmore surfaces, where the constraints are given by area, volume, and total mean curvature, assuming that the underlying conformal structures remain bounded. Furthermore, we show strong compactness of constrained Willmore surfaces under some energy threshold, proving in particular the strong compactness of minimizers of two previously studied problems.