  In this paper, we derive a new form of maximum principle for smooth functions
on a complete noncompact Riemannian manifold $M$ for which there exists a
bounded vector field $X$ such that $\langle\nabla f,X\rangle\geq 0$ on $M$ and
$\mathrm{div} X\geq af$ outside a suitable compact subset} of $M$, for some
constant $a>0$, under the assumption that $M$ has either polynomial or
exponential volume growth. We then use it to obtain some straightforward
applications to smooth functions and, more interestingly, to Bernstein-type
results for hypersurfaces immersed into a Riemannian manifold endowed with a
Killing vector field, as well as to some results on the existence and size of
minimal submanifolds immersed into a Riemannian manifold endowed with a
conformal vector field.
