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

Simon Blatt

doi:10.1007/s00028-019-00483-y

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 8π+min16πλ13/3λ22,8π\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} 8\\pi +\\min \\left\\{ \\left( 16 \\pi \\lambda _1^3\\right) \\bigg /\\left( 3\\lambda _2^2\\right) , 8\\pi \\right\\} \\end{aligned}$$\\end{document}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 4pi , i.e., close to the Willmore energy of a round sphere, converge to round points in finite time.

Highlights

3λ22, 8π and positive volume evolve singularities in finite time
Denotes the mean curvature and μ f the surface measure on Σ induced by f
The constant H0 ∈ R is called spontaneous curvature and vol( f ) denotes the signed inclosed volume given by vol( f ) =

Summary

Review of the results of McCoy and Wheeler

In the pioneering paper [17], McCoy and Wheeler completely classified all critical immersions of the functional of complete surface without boundary under the assumption that. (2.1) in a certain way says that the distance of f0 from parametrizing a round sphere is small. [17, Theorem 1] There is an absolute constant ε0 > 0 such that the following holds: Suppose that f : Σ → R3 is a smooth properly immersed complete surface without boundary and. If λ1 and λ2 or both nonnegative and at least one of them is different from zero, the solution to (1.5) sub-converges to a round point in finite time. W λ1,λ2 ( f0) ≤ 4π + ε0 sub-converge to a round point under the evolution Eq (1.5) One of the main ingredients to the proof of their theorem above is the highly nontrivial fact that under the condition (2.1) we have. Theorem 1.2 reduces the gap between the existence or better nonexistence of critical points in Theorem 2.1 and the existence of finite time singularities in Theorem 2.2

Existence of finite time singularities: an approach based on scaling

Blowup analysis of finite time singularities