Journal of the
Korean Mathematical Society JKMS

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 $\beta$ 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.

Keywords: $H^{\beta}$-curvature flow, horosphere, convex hypersurface, hyperbolic space

MSC numbers: Primary 53C44, 35K55, 58J35, 35B40