Evolution of complete noncompact graphs by powers of curvature function

2019/01/14 by Li, Guanghan, Lv, Yusha
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.04099

Abstract

This paper concerns the evolution of complete noncompact locally uniformly convex hypersurface in Euclidean space by curvature flow, for which the normal speed Φ is given by a power β≥ 1 of a monotone symmetric and homogeneous of degree one function F of the principal curvatures. Under the assumption that F is inverse concave and its dual function approaches zero on the boundary of positive cone, we prove that the complete smooth strictly convex solution exists and remains a graph until the maximal time of existence. In particular, if F=K^s/nG^1-s for any s∈(0, 1], where G is a homogeneous of degree one, increasing in each argument and inverse concave curvature function, we prove that the complete noncompact smooth strictly convex solution exists and remains a graph for all times.

Related

• Convex solutions to the power-of-mean curvature flow
• Contracting convex hypersurfaces by functions of the mean curvature
• Evolution of non-compact hypersurfaces by inverse mean curvature
• Noncollapsing for Curvature Flows with Inhomogeneous Speeds
• Entire Graphs Evolving by Powers of the Mean Curvature