Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. the first part consider volume preserving by a family curvature functions including positive powers $k$-th mean curvatures with $k=1,\ldots,n$, $p$-th power sums $S\_p$ $p > 0$. We that if initial hypersurface $M\_0$ is smooth closed has sectional curvatures, then solution Mt flow for any time $t 0$, exists all converges geodesic sphere exponentially topology. The convergence result can be used show certain Alexandrov–Fenchel quermassintegral inequalities, known previously horospherically convex hypersurfaces, also hold under weaker condition curvature. second strictly space speed given smooth, symmetric, increasing degree one homogeneous function $f$ shifted principal $\lambda\_i=\kappa\_i-1$, plus global term chosen impose constraint on quermassintegrals enclosed domain, where assumed satisfy derivatives. convex, As applications result, new rigidity theorem Weingarten class type inequalities space.