Finite time singularities for the locally constrained Willmore flow of surfaces

TL;DR

This paper investigates the finite-time singularity formation in the gradient flow of a surface energy functional combining Willmore energy, surface area, and volume, revealing conditions under which surfaces contract to points.

Contribution

It introduces new analytical tools for studying the flow, including a concentration-compactness alternative and interior estimates, and demonstrates finite-time contraction to round points for small energy initial data.

Findings

Flow contracts to a round point in finite time for small energy initial data.

Preservation of embeddedness under certain conditions.

Finite maximal existence time for the flow.

Abstract

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.