Divergence of [formula omitted] vector fields and nontrivial minimal sets on 2-manifolds

Abstract

We prove a Bendixson–Dulac type criterion for the nonexistence of nontrivial compact minimal sets of C 1 vector fields on orientable 2-manifolds. As a corollary we get that the divergence with respect to any volume 2-form of such a vector field must vanish at some point of any nontrivial compact minimal set. We also prove that all the nontrivial compact minimal sets of a C 1 vector field on an orientable 2-manifold are contained in the vanishing set of any inverse integrating factor. From this we get that if a C 1 vector field on an orientable 2-manifold has a nontrivial compact minimal set, then an infinitesimal symmetry is inessential on the minimal set.