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This work investigates the formation of singularities under the steepestdescent $L^2$-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 withoutboundary 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 thisflow. If $\lambda_1 >0$ and $\lambda_2 > 0$, embedded closed surfaces withenergy less than $$8\pi+\min\{(16 \pi \lambda_1^3)/(3\lambda_2^2), 8\pi\}$$ andpositive volume evolve singularities in finite time. If in this case theinitial 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 discusssimilar results for the case that $\lambda_2$ is negative.

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