Energy Quantization of Willmore surfaces at the boundary of the Moduli Space

TL;DR

This paper proves an energy quantization result for Willmore surfaces degenerating in moduli space, introducing a new residue to quantify energy loss and establishing compactness under certain conditions.

Contribution

It introduces a novel residue concept to measure energy loss in degenerating Willmore surfaces and proves compactness results for bounded energy and conformal class.

Findings

Energy quantization for degenerating Willmore surfaces

Introduction of a new residue quantifying energy loss

Compactness of Willmore immersions with bounded energy below 12π

Abstract

We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. we notably exhibit a new residue which quantifies the potential loss of energy in collar regions. Thanks to these residues, we also prove compactness of Willmore immersion with bounded conformal class and energy below $12\pi$.