Horospherical convex hypersurfaces contracting of the hyperbolic space by functions of the mean curvature

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 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.