  In this article, under mild constraints on the sectional curvature, we
exploit a divergence formula for symmetric endomorphisms to deduce a general
Poincar\'e type inequality. We apply such inequality to higher-order mean
curvature of hypersurfaces of space forms and Einstein manifolds, to obtain
several isoperimetric inequalities, as well as rigidity results for complete
r-minimal hypersurfaces satisfying a suitable decay of the second fundamental
form at infinity. Furthermore, using these techniques, we prove flatness and
non-existence results for self-similar solutions to a large class of fully
nonlinear curvature flows.
