  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 $\mathbb{R}^m\times S^{n-m}_{\sqrt{2(n-m)(1-t)}}$, $t<1$.
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.
