  This is a revised version (minor changes and a deeper insight in the positive
curvature case).
  We prove some Caccioppoli's inequalities for the traceless part of the second
fundamental form of a complete, noncompact, finite index, constant mean
curvature hypersurface of a Riemannian manifold, satisfying some curvature
conditions. This allows us to unify and clarify many results scattered in the
literature and to obtain some new results. For example, we prove that there is
no stable, complete, noncompact hypersurface in ${\mathbb R}^{n+1},$ $n\leq 5,$
with constant mean curvature $H\not=0,$ provided that, for suitable $p,$ the
$L^p$-norm of the traceless part of second fundamental form satisfies some
growth condition.
