Passing through nondegenerate singularities in mean curvature flows

TL;DR

This paper analyzes nondegenerate cylindrical singularities in mean curvature flow, proving their isolated nature and describing how the flow's topology changes through these singularities, akin to performing surgery.

Contribution

It introduces a new $L^2$-distance monotonicity formula and characterizes the topology change during flow passing through singularities, linking it to Morse theory.

Findings

Singularities are isolated in spacetime.

Topology change matches level set changes near Morse critical points.

New $L^2$-distance monotonicity formula established.

Abstract

In this paper, we study the properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing through the singularities. Particularly, the topology change agrees with the level sets change near a critical point of a Morse function, which is the same as performing surgery. The proof is based on a new $L^2$-distance monotonicity formula, which allows us to derive a discrete almost monotonicity of the ``decay order", a discrete mean curvature flow analog to Almgren's frequency function.