TY - JOUR

T1 - Interior estimates for hypersurfaces evolving by their k-th Weingarten curvature, and some applications

AU - Holland, James N.

N1 - Publisher Copyright:
Indiana University Mathematics Journal © 2014.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Curvature flow

KW - Interior estimates for nonlinear parabolic equations

KW - Prescribed curvature

KW - Weingarten curvature

UR - http://www.scopus.com/inward/record.url?scp=84988428126&partnerID=8YFLogxK

U2 - 10.1512/iumj.2014.63.5384

DO - 10.1512/iumj.2014.63.5384

M3 - Article

SN - 0022-2518

VL - 63

SP - 1281

EP - 1310

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

IS - 5

ER -