Interior estimates for hypersurfaces evolving by their k-th Weingarten curvature, and some applications

In this paper, we derive gradient and curvature estimates (interior in both space and time) for hypersurfaces evolving under the action of their k-th Weingarten curvature, provided that they can be locally parameterized as a graph. We do this by studying the associated parabolic PDE. This is the first time such estimates have been obtained for k-curvature flow, excepting the mean curvature case, where the analogous results are in [3] and [4]. As an application of these estimates, we obtain global existence results for the Cauchy problem of k-curvature flow of entire hypersurfaces under very weak regularity assumptions on the initial data. We also demonstrate that if the initial entire hypersurface is asymptotic to a cone in some weak sense, then the associated solution, after rescaling, will converge to a self-similar solution which evolves homothetically. These results are the first time the k-curvature flow of entire hypersurfaces has been investigated for k ≠ 1, and they generalize all the key results for the mean curvature case in [3], [4], and [13] to this more general setting. The results are new even in the case of Gauss curvature flow.