  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.
