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

Abstract

We prove a quantitative reverse isoperimetric inequality for embedded surfaces with Willmore energy bounded away from $8\pi$. We use this result to analyze the negative $L^2$ gradient flow of the Willmore energy plus a positive multiple of the inclosed volume. We show that initial surfaces of Willmore energy less than $8\pi$ with positive inclosed volume converge to a round point in finite or infinite time.