Finite time singularities for the locally constrained Willmore flow of surfaces

In this paper we study the steepest descent L2-gradient flow of the functional Wλ1,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.