Finite time singularities for the locally constrained Willmore flow of surfaces

2012/01/22 by McCoy, James, Wheeler, Glen
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1201.4541

Abstract

In this paper we study the steepest descent L²-gradient flow of the functional \SW_λ1,λ2, which is the the sum of the Willmore energy, λ₁-weighted surface area, and λ₂-weighted enclosed volume, for surfaces immersed in \R³. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our first results are a concentration-compactness alternative and interior estimates for the flow. For initial data with small energy, we prove preservation of embeddedness, and by directly estimating the Euler-Lagrange operator from below in L² we obtain that the maximal time of existence is finite. Combining this result with the analysis of a suitable blowup allows us to show that for such initial data the flow contracts to a round point in finite time.

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