A note on singularities in finite time for the constrained Willmore flow

Abstract

This work investigates the formation of singularities under the steepest descent $L^2$-gradient flow of the functional ${\mathcal{W}}_{{\lambda }_1,{\lambda }_2}$, the sum of the Willmore energy, ${\lambda }_1$ times the area, and ${\lambda }_2$ times the signed volume of an immersed closed surface without boundary in ${\mathbb{R}}^3$. We show that in the case that ${\lambda }_1>1$ and ${\lambda }_2=0$ any immersion develops singularities in finite time under this flow. If ${\lambda }_1>0$ and ${\lambda }_2>0$, embedded closed surfaces with energy less than $$8\pi +\min \{(16\pi {\lambda }_1^3)/(3{\lambda }_2^2),8\pi \}$$ and positive volume evolve singularities in finite time. If in this case the initial surface is a topological sphere and the initial energy is less than $8\pi$, the flow shrinks to a round point in finite time. We furthermore discuss similar results for the case that ${\lambda }_2$ is negative.