  We prove differential Harnack inequalities for flows of strictly convex
hypersurfaces by powers $p$, $0<p<1$, of the mean curvature in Einstein
manifolds with a positive lower bound on the sectional curvature. We assume
that this lower bound is sufficiently large compared to the derivatives of the
curvature tensor of the ambient space and that the mean curvature of the
initial hypersurface is sufficiently large compared to the ambient geometry. We
also obtain some new Harnack inequalities for more general curvature flows in
the sphere, as well as a monotonicity estimate for the mean curvature flow in
non-negatively curved, locally symmetric spaces.
