  In this paper, we study flows of hypersurfaces in hyperbolic space, and apply
them to prove geometric inequalities. In the first part of the paper, we
consider volume preserving flows by a family of curvature functions including
positive powers of $k$-th mean curvatures with $k=1,\cdots,n$, and positive
powers of $p$-th power sums $S_p$ with $p>0$. We prove that if the initial
hypersurface $M_0$ is smooth and closed and has positive sectional curvatures,
then the solution $M_t$ of the flow has positive sectional curvature for any
time $t>0$, exists for all time and converges to a geodesic sphere
exponentially in the smooth topology. The convergence result can be used to
show that certain Alexandrov-Fenchel quermassintegral inequalities, known
previously for horospherically convex hypersurfaces, also hold under the weaker
condition of positive sectional curvature.
  In the second part of this paper, we study curvature flows for strictly
horospherically convex hypersurfaces in hyperbolic space with speed given by a
smooth, symmetric, increasing and homogeneous degree one function $f$ of the
shifted principal curvatures $\lambda_i=\kappa_i-1$, plus a global term chosen
to impose a constraint on the quermassintegrals of the enclosed domain, where
$f$ is assumed to satisfy a certain condition on the second derivatives. We
prove that if the initial hypersurface is smooth, closed and strictly
horospherically convex, then the solution of the flow exists for all time and
converges to a geodesic sphere exponentially in the smooth topology. As
applications of the convergence result, we prove a new rigidity theorem on
smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of
Alexandrov-Fenchel type inequalities for smooth horospherically convex
hypersurfaces in hyperbolic space.
