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

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