Energy quantization for Willmore surfaces with bounded index

TL;DR

This paper establishes an energy quantization principle for Willmore surfaces with bounded index, analyzing their behavior via the conformal Gauss map, especially in degenerating Riemann surfaces.

Contribution

It introduces a novel approach using the conformal Gauss map to prove energy quantization for Willmore surfaces, including degenerating cases.

Findings

Conformal Gauss map converges to a light-like geodesic in neck regions.

Energy quantization holds regardless of Riemann surface degeneration.

Provides a new perspective on Willmore surface analysis through De Sitter space.

Abstract

We prove an energy quantization result for Willmore surfaces with bounded index, whether the underlying Riemann surfaces degenerates in the moduli space or not. To do so, we translate the question on the conformal Gauss map's point of view. In particular, we prove that in a neck or a collar region, the conformal Gauss map converges to a light-like geodesic in the De Sitter space.