Approximation of mean curvature flow with generic singularities by smooth flows with surgery

TL;DR

This paper develops a method to approximate weak mean curvature flows with singularities using smooth flows with surgery, combining recent theoretical advances and barrier techniques, and applies it to improve entropy bounds related to the Schoenflies conjecture.

Contribution

The paper introduces a new approach to approximate weak mean curvature flows with singularities by smooth flows with surgery, leveraging recent canonical neighborhood results.

Findings

Constructed smooth flows with surgery approximating weak flows with spherical and neck-pinch singularities.

Established canonical neighborhoods for such singularities.

Improved entropy bounds on the low-entropy Schoenflies conjecture.

Abstract

We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We conclude by improving the entropy bound on the low-entropy Schoenflies conjecture.