Revisiting generic mean curvature flow in $\mathbb{R}^3$

TL;DR

This paper advances the understanding of mean curvature flow in three-dimensional space by proving that a density-drop theorem combined with recent multiplicity-one results suffices to resolve Huisken's genericity conjecture, simplifying previous approaches.

Contribution

It demonstrates that the resolution of Huisken's conjecture can be achieved without relying on the strict genus drop theorem, using a new combination of existing theorems.

Findings

Proves a short density-drop theorem for mean curvature flow.

Shows that multiplicity-one results suffice to resolve Huisken's conjecture.

Simplifies previous proofs by removing the need for the genus drop theorem.

Abstract

Bamler--Kleiner recently proved a multiplicity-one theorem for mean curvature flow in R^3 and combined it with the authors' work on generic mean curvature flows to fully resolve Huisken's genericity conjecture. In this paper we show that a short density-drop theorem plus the Bamler--Kleiner multiplicity-one theorem for tangent flows at the first nongeneric singular time suffice to resolve Huisken's conjecture -- without relying on the strict genus drop theorem for one-sided ancient flows previously established by the authors.