On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space

Abstract

In this paper, we study a class of flows of closed, star-shaped hypersurfaces in hyperbolic space $\mathbb{H}^{n+1}$ with speed $(\sinh r)^{α/β} σ_{k}^{{1}/β}$, where $σ_{k}$ is the $k$-th elementary symmetric polynomial of the principal curvatures, $α$, $ β$ are positive constants and $r$ is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of $k$, $α$ and $ β$. When $k = 1 , α> 1 + β$, and the initial hypersurface is mean convex, we prove that the mean convex solution to the flow for $ k=1 $ exists for all time and converges smoothly to a sphere. When $1\leq k \leq n, α> k+β$, and the initial hypersurface is uniformly convex, we prove that the uniformly convex solution to the flow exists for all time and converges smoothly to a sphere. In particular, we generalize Li-Sheng-Wang's results from Euclidean space to hyperbolic space.