Willmore Flow of Complete Surfaces

TL;DR

This paper studies the evolution of complete surfaces under the Willmore flow, proving short-term existence, uniqueness, and characterizing low-energy solutions as planes, with convergence results under certain conditions.

Contribution

It extends previous results by establishing short-time existence and uniqueness for the Willmore flow of complete surfaces with bounded geometry, and characterizes low-energy solutions as planes.

Findings

Complete Willmore surfaces with low energy are planes.

Willmore flow with low initial energy converges to a plane.

Established short-time existence and uniqueness for the flow.

Abstract

We consider the Willmore flow equation for complete, properly immersed surfaces in Rn. Given bounded geometry on the initial surface, we extend the result by Kuwert and Sch\"atzle in 2002 and prove short time existence and uniqueness of the Willmore flow. We also show that a complete Willmore surface with low Willmore energy must be a plane, and that a Willmore flow with low initial energy and Euclidean volume growth must converge smoothly to a plane.