Revisiting generic mean curvature flow in $\mathbb{R}^3$
Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, Felix Schulze
TL;DR
This paper addresses Huisken's genericity conjecture for mean curvature flow in $\mathbb{R}^3$ by replacing the prior genus-drop strategy with a density-drop approach that leverages Bamler--Kleiner's multiplicity-one tangent-flow results. It shows that a short density-drop theorem, together with a detailed understanding of tangent flows at the first nongeneric singular time, forces all nearby cyclic unit-regular Brakke flows to develop only multiplicity-one spherical or cylindrical singularities. As a consequence, under genus/entropy bounds the set of nongeneric singularities is finite, enabling an open-dense collection of initial data for which Huisken's conjecture holds and yielding a shorter, global proof. The analysis combines an entropy-gap for shrinkers, shrinker compactness, Frankel-type rigidity for $F$-stationary varifolds, and the Ilmanen–Bamler–Kleiner tangent-flow structure theorem to achieve the result.
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.