Communications in Analysis and Geometry

Volume 24 (2016)

Number 4

Finite time singularities for the locally constrained Willmore flow of surfaces

Pages: 843 – 886

DOI: https://dx.doi.org/10.4310/CAG.2016.v24.n4.a7

Authors

James McCoy (Institute for Mathematics and Applied Statistics, University of Wollongong, NSW, Australia)

Glen Wheeler (Institut für Analysis und Numerik, Otto-von-Guericke-Universität, Magdeburg, Germany; and Institute for Mathematics and Applied Statistics, University of Wollongong, NSW, Australia)

Abstract

In this paper we study the steepest descent $L^2$-gradient flow of the functional $W_{\lambda_1, \lambda_2}$, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-weighted enclosed volume, for surfaces immersed in $\mathbb{R}^3$. 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 $L^2$ 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.

Keywords

global differential geometry, fourth order, geometric analysis, parabolic partial differential equations

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Published 3 November 2016