The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere

TL;DR

This paper extends the classification of branched Willmore spheres in 3- and 4-spheres, linking them to minimal surfaces and analyzing their regularity and min-max properties.

Contribution

It generalizes Bryant's classification to branched spheres, provides new regularity results, and establishes quantization of min-max widths in dimensions 3 and 4.

Findings

Branched Willmore spheres in S^3 are inverse stereographic projections of minimal surfaces with finite total curvature.

Improved C^{1,1} regularity of the unit normal at branch points.

Min-max widths for Willmore spheres are integer multiples of 4π.

Abstract

We extend the classification of Robert Bryant of Willmore spheres in $S^3$ to variational branched Willmore spheres $S^3$ and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in $\mathbb{R}^3$ and vanishing flux. We also obtain a classification of variational branched Willmore spheres in $S^4$, generalising a theorem of Seb\'{a}stian Montiel. As a result of our asymptotic analysis at branch points, we obtain an improved $C^{1,1}$ regularity of the unit normal of variational branched Willmore surfaces in arbitrary codimension. We also prove that the width of Willmore sphere min-max procedures in dimension $3$ and $4$, such as the sphere eversion, is an integer multiple of $4\pi$.