A note on Alexandrov immersed mean curvature flow

TL;DR

This paper shows that Alexandrov immersed surfaces maintain their property under mean curvature flow and adapts existing techniques for embedded flows to this broader class, including modifications for flow with surgery.

Contribution

It extends mean curvature flow methods to Alexandrov immersed surfaces, including noncollapsing, gradient estimates, and flow with surgery adaptations.

Findings

Alexandrov immersed property is preserved during mean curvature flow

Existing embedded flow techniques are applicable to Alexandrov immersed flows

Modifications enable mean curvature flow with surgery for Alexandrov immersed surfaces

Abstract

We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle--Huisken to allow for mean curvature flow with surgery for the Alexandrov immersed, $2$-dimensional setting.