2021/11/19 by Hong, Fang
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2111.10170
In this paper, we study a class of flows of closed, star-shaped hypersurfaces in hyperbolic space ℍn+1 with speed (\sinh r)α/β σk^1/β, where σk 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≤ k ≤ 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.