A note on singularities in finite time for the \(L^{2}\) gradient flow of the Helfrich functional

Abstract

This work investigates the formation of singularities under the steepest descent \(L^2\)-gradient flow of \({{\,\mathrm{{ W}}\,}}_{\lambda _1, \lambda _2}\) with zero spontaneous curvature, i.e., 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

$$\begin{aligned} 8\pi +\min \left\{ \left( 16 \pi \lambda _1^3\right) \bigg /\left( 3\lambda _2^2\right) , 8\pi \right\} \end{aligned}$$

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. These results strengthen the ones of McCoy and Wheeler (Commun Anal Geom 24(4):843–886, 2016). For \(\lambda _1 >0\) and \(\lambda _2 \ge 0\), they showed that embedded closed spheres with positive volume and energy close to \(4\pi \), i.e., close to the Willmore energy of a round sphere, converge to round points in finite time.

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Correspondence to Simon Blatt.

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The author was supported through FWF Grant P 29487 “The gradient flow of curvature energies”.

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Blatt, S. A note on singularities in finite time for the \(L^{2}\) gradient flow of the Helfrich functional. J. Evol. Equ. 19, 463–477 (2019). https://doi.org/10.1007/s00028-019-00483-y

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