Finite time singularities for the locally constrained Willmore flow of surfaces

Details Finite time singularities for the locally constrained Willmore flow of surfaces Finite time singularities for the locally constrained Willmore flow of surfaces

2012-01-22

arxiv.org/abs/1201.4541v1

Mathematics

Differential Geometry

31 pages

Scientific paper

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

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