  This paper gives some examples of hypersurfaces $\phi_t(M^n)$ evolving in
time with speed determined by functions of the normal curvatures in an
$(n+1)$-dimensional hyperbolic manifold; we emphasize the case of flow by
harmonic mean curvature. The examples converge to a totally geodesic
submanifold of any dimension from 1 to $n$, and include cases which exist for
infinite time. Convergence to a point was studied by Andrews, and only occurs
in finite time. For dimension $n=2,$ the destiny of any harmonic mean curvature
flow is strongly influenced by the genus of the surface $M^2$.
