On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space

TL;DR

This paper investigates a class of fully nonlinear curvature flows of star-shaped hypersurfaces in hyperbolic space, establishing conditions for long-term existence and smooth convergence to spheres, thus extending Euclidean results to hyperbolic geometry.

Contribution

It generalizes previous Euclidean space results on curvature flows to hyperbolic space, providing new convergence theorems for specific flow conditions.

Findings

Flow solutions exist for all time under certain convexity conditions.

Solutions converge smoothly to spheres when conditions on parameters are met.

Extends Li-Sheng-Wang's Euclidean results to 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.