Higher Regularity of Weak Limits of Willmore Immersions II

TL;DR

This paper establishes a removability result for singularities of Willmore surfaces in arbitrary codimension, constraining the types of minimal surfaces that can appear as bubbles in Willmore min-max problems.

Contribution

It provides a new removability theorem for singularities of Willmore surfaces and limits the minimal surfaces that can arise in min-max constructions.

Findings

Only three of twelve non-planar minimal surfaces with total curvature > -12π can appear as bubbles.

The removability result applies in arbitrary codimension.

Constraints on bubble formation in Willmore min-max problems.

Abstract

We obtain in arbitrary codimension a removability result on the order of singularity of Willmore surfaces realising the width of Willmore min-max problems on spheres. As a consequence, out of the twelve families of non-planar minimal surfaces in $\mathbb{R}^3$ of total curvature greater than $-12\pi$, only three of them may occur as conformal images of bubbles in Willmore min-max problems.