Finite time singularities for the locally constrained Willmore flow of surfaces

Title: Finite time singularities for the locally constrained Willmore flow of surfaces
Creator: Mccoy, James; Wheeler, Glen
Relation: Communications in Analysis and Geometry Vol. 24, Issue 4, p. 843-886
Publisher Link: http://dx.doi.org/10.4310/CAG.2016.v24.n4.a7
Publisher: International Press
Resource Type: journal article
Date: 2016
Description: In this paper we study the steepest descent L²-gradient flow of the functional W_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.
Subject: global differential geometry; fourth order; geometric analysis; parabolic partial differential equations
Identifier: http://hdl.handle.net/1959.13/1437146
Identifier: uon:40250
Identifier: ISSN:1019-8385
Language: eng
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