Analysis of Constrained Willmore Surfaces

TL;DR

This paper investigates the regularity and local behavior of constrained Willmore surfaces, providing conditions for smoothness and detailed asymptotic expansions near regular and branch points.

Contribution

It introduces new local asymptotic expansion techniques and explicit point removability conditions for constrained Willmore immersions in arbitrary codimension.

Findings

Derived local asymptotic expansions in terms of circulation integrals.

Established explicit conditions for point removability and smoothness.

Applied results to Willmore and parallel mean curvature immersions.

Abstract

This paper studies the regularity of constrained Willmore immersions into $\R^{m\ge3}$ locally around both "regular" points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion, its first, and its second derivatives, given in terms of residues which are computed as circulation integrals. We deduce explicit "point removability" conditions ensuring that the immersion is smooth. Our results apply in particular to Willmore immersions and to parallel mean curvature immersions in any codimension.