A class of curvature flows expanded by support function and curvature function in the Euclidean space and hyperbolic space

Abstract

In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space Rn+1 with speed uαf−β, where u is the support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For α⩽0<β⩽1−α, we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt [16] and Urbas [40] can be recovered by putting α=0 and β=1 in our first result. If the initial hypersurface is convex, this is our previous work [11]. If α⩽0<β<1−α and the ambient space is hyperbolic space Hn+1, we prove that the flow ∂X∂t=(uαf−β−ηu)ν has a longtime existence and smooth convergence to a coordinate slice. The flow in Hn+1 is equivalent (up to an isomorphism) to a re-parametrization of the original flow in Rn+1 case. Finally, we find a family of monotone quantities along the flows in Rn+1. As applications, we give a new proof of a family of inequalities involving the weighted integral of kth elementary symmetric function for k-convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in [20].