Planarity and convexity for pinched ancient solutions of mean curvature flow

We prove a parabolically scale-invariant variation of the planarity estimate in \cite{Na22} for higher codimension mean curvature flow, borrowing ideas from work of Brendle–Huisken–Sinestrari \cite{BHS}. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.