A monotonicity formula for mean curvature flow with surgery

TL;DR

This paper introduces a new monotonicity formula for mean curvature flow with surgery, which helps determine when surgeries are unnecessary, aiding the understanding of the flow's long-term behavior in three-manifolds.

Contribution

It presents a novel monotonicity formula that accounts for surgeries and shows conditions under which surgeries can be avoided, advancing the analysis of mean curvature flow.

Findings

The new formula includes an extra mean curvature term.

Flow close to smooth in geometric measure theory is surgery-free.

Applications to long-term behavior in Riemannian three-manifolds.

Abstract

We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is, in fact, free of surgeries. This result has applications to the longtime behavior of mean curvature flow with surgery in Riemannian three-manifolds.