The blowdown of ancient noncollapsed mean curvature flows

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, $\lim_{λ\to 0} λK_{t_0}$, 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.