A general pinching principle 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 (J Differ Geom 20(1): 237-266, 1984), the convexity estimates of Huisken-Sinestrari (Acta Math 183(1): 45-70, 1999) and the cylindrical estimate of Huisken-Sinestrari (Invent Math 175(1): 137-221, 2009; see also Andrews and Langford in Anal PDE 7(5): 1091-1107, 2014; Huisken and Sinestrari in J Differ Geom 101(2): 267287, 2015). 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 R-m x S-root 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 (Brendle in Invent Math 202(1): 217-237, 2015; Haslhofer and Kleiner in Int Math Res Not 15: 6558-6561, 2015; Langford in Proc Am Math Soc 143(12): 5395-5398, 2015). Making use of a recent idea of Huisken-Sinestrari (2015), 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 strengthen recent characterizations of the shrinking sphere due to Huisken-Sinestrari (2015) and Haslhofer-Hershkovits