# Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions

## TL;DR

This paper establishes sharp one-sided curvature estimates for fully nonlinear curvature flows, providing new tools for analyzing ancient solutions and characterizing shrinking spheres in geometric evolution equations.

## Contribution

It introduces novel sharp curvature pinching estimates for nonlinear flows and applies these to classify ancient solutions, including characterizations of shrinking spheres.

## Key findings

- Sharp estimates for principal curvatures and inscribed curvature
- Characterization of shrinking spheres among ancient solutions
- Extension of estimates to flows by convex and concave speeds

## Abstract

We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the largest principal curvature and the inscribed curvature ('cylindrical estimates') for flows by concave speeds and a sharp estimate for the exscribed curvature for flows by convex speeds. Making use of a recent idea of Huisken and Sinestrari, we then obtain corresponding estimates for ancient solutions. In particular, this leads to various characterisations of the shrinking sphere amongst ancient solutions of these flows.