IUMJ

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

Authors: James Holland

Issue: Volume 63 (2014), Issue 5, 1281-1310

Abstract:

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 [K. Ecker and G. Huisken, \textit{Mean curvature evolution of entire graphs}, Ann. of Math. (2) \textbf{130} (1989), no. 3, 453--471] and [K. Ecker and G. Huisken, \textit{Interior estimates for hypersurfaces moving by mean curvature}, Invent. Math. \textbf{105} (1991), no. 3, 547--569]. 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 $k \neq 1$, and they generalize all the key results for the mean curvature case in [K. Ecker and G. Huisken, \textit{Mean curvature evolution of entire graphs}, Ann. of Math. (2) \textbf{130} (1989), no. 3, 453--471], [K. Ecker and G. Huisken, \textit{Interior estimates for hypersurfaces moving by mean curvature}, Invent. Math. \textbf{105} (1991), no. 3, 547--569], and [N. Stavrou, \textit{Selfsimilar solutions to the mean curvature flow}, J. Reine Angew. Math. \textbf{499} (1998), 189--198] to this more general setting. The results are new even in the case of Gauss curvature flow.