This article is about contraction by fully nonlinear curvature flows of convex hypersurfaces. As with previously considered flows, including the quasilinear mean curvature flow and a broad class of fully nonlinear curvature flows, solutions exist for a finite time and contract to a point. As with some other flows, including the mean curvature flow, a smaller class of fully nonlinear flows and the Gauss curvature flow for surfaces, under a suitable rescaling the solutions converge exponentially to spheres.
We provide an introduction to the study of curvature flow of hypersurfaces with extensive references to most of the main areas of study and applications to other areas of mathematics and beyond. The main points of interest in this particular work are the allowance of nonsmooth initial data and its consequences and that the only second derivative requirement on the speed is weaker than a requirement of convexity or concavity. We outline new results in both cases of smooth and nonsmooth initial data. These results are joint work with Ben Andrews and Zheng Yu.