Andrews, BenChen, Xuzhong2021-06-032021-06-030075-4102http://hdl.handle.net/1885/236350We study the evolution of compact convex hypersurfaces in hyperbolic space ℍn+1, with normal speed governed by the curvature. We concentrate mostly on the case of surfaces, and show that under a large class of natural flows, any compact initial surface with Gauss curvature greater than 1 produces a solution which converges to a point in finite time, and becomes spherical as the final time is approached. We also consider the higher-dimensional case, and show that under the mean curvature flow a similar result holds if the initial hypersurface is compact with positive Ricci curvature.The first author was partly supported by Discovery Projects grant DP1200097 of the Australian Research Council. The second author was partly supported by National Natural Science Foundation of China under grants 11271132, 11071212 and 11131007. The authors are grateful for the hospitality of the Mathematical Sciences Center of Tsinghua University where the research was carried outapplication/pdfen-AU© De Gruyter 2015201710.1515/crelle-2014-01212020-11-23