Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

In a recent article, a localization of the Huisken--Stampacchia iteration method was developed, and used to establish localizations of the well-known "umbilic", "convexity" and "cylindrical" estimates for hypersurfaces evolving in Euclidean space by mean curvature flow. Here, we adapt the methods developed there to treat more general (fully nonlinear) flows, establishing localizations of asymptotically sharp curvature pinching estimates for hypersurfaces evolving by one-homogeneous functions of curvature under very general conditions. We also briefly describe how the method can be adapted to treat the deformation of hypersurfaces in curved ambient spaces (by suitable speed functions), which is fundamental for many important applications of such flows.