The blowdown of ancient noncollapsed mean curvature flows

TL;DR

This paper proves that ancient noncollapsed mean curvature flows in higher dimensions have blowdowns of at most (n-2) dimensions, refining previous results and advancing understanding of singularity structures.

Contribution

It establishes that the blowdown of such flows is at most (n-2) dimensional, extending previous results and applying fine cylindrical analysis techniques.

Findings

Blowdowns are at most (n-2) dimensional for general ancient noncollapsed flows.

In the k-convex case, blowdowns are at most (k-2) dimensional.

Results generalize recent work to higher dimensions and aid classification of singularities.

Abstract

In this paper, we consider ancient noncollapsed mean curvature flows $M_t=\partial K_t\subset \mathbb{R}^{n+1}$ that do not split off a line. It follows from general theory that the blowdown of any time-slice, $\lim_{\lambda \to 0} \lambda K_{t_0}$, is at most $n-1$ dimensional. Here, we show that the blowdown is in fact at most $n-2$ dimensional. Our proof is based on fine cylindrical analysis, which generalizes the fine neck analysis that played a key role in many recent papers. Moreover, we show that in the uniformly $k$-convex case, the blowdown is at most $k-2$ dimensional. This generalizes recent results from Choi-Haslhofer-Hershkovits to higher dimensions, and also has some applications towards the classification problem for singularities in 3-convex mean curvature flow.