Interior estimates for hypersurfaces evolving by their k-th Weingarten curvature and some applications James Holland 53C4435K55Curvature FlowPrescribed CurvatureWeingarten CurvatureInterior Estimates for Nonlinear Parabolic Equations 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. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5384 10.1512/iumj.2014.63.5384 en Indiana Univ. Math. J. 63 (2014) 1281 - 1310 state-of-the-art mathematics http://iumj.org/access/