# Asymptotic estimates for the Willmore flow with small energy

## TL;DR

This paper provides asymptotic estimates for the Willmore flow with small initial energy, demonstrating stability of geometric quantities and recovering known rigidity and isoperimetric estimates through new methods.

## Contribution

It establishes stability estimates for geometric quantities under the Willmore flow with small energy, offering alternative proofs for known results.

## Key findings

- Stability estimates for barycenter and quadratic moment.
- Bounds for enclosed volume and mean curvature in codimension one.
- Recovery of existing rigidity and isoperimetric estimates using new methods.

## Abstract

Kuwert and Sch\"atzle showed in 2001 that the Willmore flow converges to a standard round sphere, if the initial energy is small. In this situation, we prove stability estimates for the barycenter and the quadratic moment of the surface. Moreover, in codimension one we obtain stability bounds for the enclosed volume and averaged mean curvature. As direct applications, we recover a quasi-rigidity estimate due to De Lellis and M\"uller (2006) and an estimate for the isoperimetric deficit by R\"oger and Sch\"atzle (2012), whose original proofs used different methods.