A Reverse Isoperimetric Inequality and its Application to the Gradient Flow of the Helfrich Functional

2020/09/25 by Blatt, Simon
#47J35 #53A05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2009.12273

Abstract

We prove a quantitative reverse isoperimetric inequality for embedded surfaces with Willmore energy bounded away from 8π. We use this result to analyze the negative L² gradient flow of the Willmore energy plus a positive multiple of the inclosed volume. We show that initial surfaces of Willmore energy less than 8π with positive inclosed volume converge to a round point in finite or infinite time.

Related

• Isoperimetric problems and lower bounds on curvature
• Control of the isoperimetric deficit by the Willmore deficit
• Finite time singularities for the locally constrained Willmore flow of surfaces
• Gradient Flow for the Willmore Functional in Riemannian Manifolds of bounded Geometry
• A strong form of the quantitative Wulff inequality