A fully nonlinear flow of three-convex hypersurfaces

We will discuss a fully nonlinear geometric flow of three-convex hypersurfaces, where the normal speed at each point of the solution is given by a concave function of the second fundamental form. Three-convexity means that at each point, the sum of the smallest three principal curvatures is positive. The flow smoothly deforms any compact three-convex initial hypersurface, preserving three-convexity, until its curvature becomes unbounded. Our main result is a convexity estimate, which says that high-curvature regions are approximately convex. Such an estimate is known to hold for mean-convex mean curvature flow, and for a large class of fully nonlinear flows where the speed function is convex. For concave speeds, previous results of this kind assume two-convexity