This paper concerns closed hypersurfaces of dimension n(≥ 2) in the hyperbolic space H n+1 κ of constant sectional curvature κ evolving in direction of its normal vector, where the speed is given by a power β(≥ 1/m) of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the Gauß curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on n, m, β and κ, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of H n+1 κ , enclosing the same volume as the initial hypersurface.2010 Mathematics Subject Classification. Primary 53C44, 35K55; Secondary, 58J35, 35B40.

Abstract. This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a positive power β of the positive mean curvature. It is shown that the flow exists on a finite maximal interval, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached.

Abstract. This paper concerns closed hypersurfaces of dimension n ≥ 2 in the hyperbolic space H n+1 κ of constant sectional curvature κ evolving in direction of its normal vector, where the speed equals a power β ≥ 1 of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and β, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in H n+1 κ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of H n+1 κ .

This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This generalizes the previous result [S. Guo, Convex hypersurfaces evolving by functions of the mean curvature, preprint (2016), arXiv:1610.08214 ] for convex hypersurfaces in the Euclidean space by the author to the setting in the hyperbolic space for the same class of flows.