A maximum principle related to volume growth and applications

Abstract

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 $$\langle \nabla f,X\rangle \ge 0$$ on M and $$\rm {div} X\ge af$$ 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 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.