Sharp one-sided curvature estimates for mean curvature flow and applications

We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of Huisken–Sinestrari and the cylindrical estimate of Huisken–Sinestrari. Namely, we show that the curvature of the solution pinches onto the convex cone generated by the curvatures of any shrinking cylinder solutions admitted by the initial data. For example, if the initial data is $(m+1)$-convex, then the curvature of the solution pinches onto the convex hull of the curvatures of the shrinking cylinders ${\mathbb{R}}^m\times S_{\sqrt{2(n-m)(1-t)}}^{n-m}$, $t<1$. In particular, this yields a sharp estimate for the largest principal curvature, which we use to obtain a new proof of a sharp estimate for the inscribed curvature for embedded solutions. Making use of a recent idea of Huisken–Sinestrari, we then obtain a series of sharp estimates for ancient solutions. In particular, we obtain a convexity estimate for ancient solutions which allows us to improve recent characterizations of the shrinking sphere due to Huisken–Sinestrari and Haslhofer–Hershkovitz.