The principal eigenvalue and maximum principle for second order elliptic operators on Riemannian manifolds

In this paper the work of Berestycki, Nirenberg and Varadhan on the maximum principle and the principal eigenvalue for second order operators on general domains is extended to Riemannian manifolds. In particular it is proved that the refined maximum principle holds for a second order elliptic operator on a manifold if and only if the principal eigenvalue is positive. (C) 1997 Academic Press.