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

2017/06/05 by Michelat, Alexis, Rivière, Tristan · 1 citation
#35J35 #35R01 #49Q10 #53A05 #53A10 #53A30 #53C42 #58E15 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1706.01405

Abstract

We extend the classification of Robert Bryant of Willmore spheres in S³ to variational branched Willmore spheres S³ and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in ℝ³ and vanishing flux. We also obtain a classification of variational branched Willmore spheres in S⁴, generalising a theorem of Sebástian 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 4π.

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• Classification of Willmore 2-spheres in Sⁿ

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