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In this paper, we derive a new form of maximum principle for smooth functionson a complete noncompact Riemannian manifold $M$ for which there exists abounded 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 someconstant $a>0$, under the assumption that $M$ has either polynomial orexponential volume growth. We then use it to obtain some straightforwardapplications to smooth functions and, more interestingly, to Bernstein-typeresults for hypersurfaces immersed into a Riemannian manifold endowed with aKilling vector field, as well as to some results on the existence and size ofminimal submanifolds immersed into a Riemannian manifold endowed with aconformal vector field.

A maximum principle related to volume growth and applications