  In this article, we will use the harmonic mean curvature flow to prove a new
class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces
in hyperbolic space in terms of total curvature, which is the integral of
Gaussian curvature on the hypersurface. We will also use the harmonic mean
curvature flow to prove a new class of geometric inequalities for
horospherically convex hypersurfaces in hyperbolic space. Using these new
Alexandrov-Fenchel type inequalities and the inverse mean curvature flow, we
obtain an Alexandrov-Fenchel inequality for strictly convex hypersurfaces in
hyperbolic space, which was previously proved for horospherically convex
hypersurfaces by Wang and Xia [44]. Finally, we use the mean curvature flow to
prove a new Heintze-Karcher type inequality for hypersurfaces with positive
Ricci curvature in hyperbolic space.
