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 $(\sinh r)^{{\alpha}/{\beta}} \sigma_{k}^{{1}/{\beta}}$, where $\sigma_{k}$ is the $k$-th elementary symmetric polynomial of the principal curvatures, $\alpha$, $ \beta $ are positive constants and $r$ is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of $k$, $\alpha$ and $ \beta $. When $k = 1 , \alpha > 1 + \beta$, 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 $1\leq k \leq n, \alpha > k+\beta$, 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.