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

TL;DR

This paper studies the formation of singularities in the constrained Willmore flow of surfaces in 3D, identifying conditions under which finite-time singularities occur or surfaces shrink to points.

Contribution

It provides new results on finite-time singularity formation for the constrained Willmore flow with various parameter regimes, including cases with positive and negative volume constraints.

Findings

Surfaces develop singularities in finite time when >1 and =0.

Embedded surfaces with energy below a certain threshold and positive volume also develop singularities.

Spheres with initial energy less than 8 shrink to round points in finite time.

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.