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 $1\le k\le n,\alpha >k+\beta$, 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.

25 pages