Volume preserving mean curvature flow in the hyperbolic space Esther Cabezas-RivasVicente Miquel 53C4453C2152A5552A20volume preserving mean curvature flowhyperbolic spaceconvex by horospheres We prove: "If $M$ is a compact hypersurface of the hyperbolic space, convex by horospheres and evolving by the volume preserving mean curvature flow, then it flows for all time, convexity by horospheres is preserved and the flow converges, exponentially, to a geodesic sphere". In addition, we show that the same conclusions about long time existence and convergence hold if $M$ is not convex by horospheres but it is close enough to a geodesic sphere. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3060 10.1512/iumj.2007.56.3060 en Indiana Univ. Math. J. 56 (2007) 2061 - 2086 state-of-the-art mathematics http://iumj.org/access/