3-circle Theorem for Willmore surfaces II--degeneration of the complex structure

TL;DR

This paper investigates the behavior of Willmore surfaces during degeneration, focusing on complex structure changes, energy loss, and the Gauss map's limit, supported by explicit examples.

Contribution

It introduces a novel analysis of Willmore surface degeneration without requiring convergence of complex structures, linking energy loss to residues and describing the Gauss map limit.

Findings

Energy loss in neck regions is quantified by residues.

The Gauss map limit is a geodesic in the Grassmannian.

Explicit examples illustrate degeneration phenomena.

Abstract

We study the compactness of Willmore surfaces without assuming the convergence of the induced complex structures. In particular, we compute the energy loss in the neck in terms of the residue and we prove that the limit of the image of the Gauss map is a geodesic in the Grassmannian $G(2,n)$ whose length can also be computed in terms of the residue. Moreover, we provide a family of explicit Willmore surfaces in $\R^3$ that illustrate the denegeration phenomenon involved in the above results.