Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold

We derive an interior planarity estimate for space-time proper mean curvature flows of $n$-dimensional submanifolds of ${\mathbb{R}}^{n+p}$, $n\ge 5$, that are uniformly quadratically pinched up to the Andrews-Baker threshold $\frac{4}{3n}$. The estimate is parabolically scale-invariant and it requires neither curvature nor entropy bounds. The proof combines Naff's gradient estimates with a localization argument in the spirit of Ecker and Huisken. In dimensions five and six, Naff's estimates alone do not give the gradient terms a sign. We bound these terms from below with an argument we developed in a previous work. As a consequence, for $n\ge 5$, and by known results for $2\le n\le 4$, every space-time proper ancient solution that is uniformly $\frac{4}{3n}$-pinched is either a static affine $n$-plane or lies in a fixed affine $(n+1)$-dimensional subspace. This replaces the finite entropy and polynomial curvature growth assumptions of Lee, Naff and Zhu by space-time properness, and improves their pinching constant in dimensions two, five and six. For closed flows, we extend Lynch's planarity estimate to a larger range of exponents.