A note on Killing fields and CMC hypersurfaces

In this note we give some sufficient conditions for a CMC-hypersurface in a Riemannian manifold N to be invariant under the 1-parameter group of isometries generated by a Killing field on N. Our main result improves on previous ones by D. Hoffman, R. Osserman, and R. Schoen and S. Fornari and J. Ripoll, and hinges on a new, simple existence theorem for a first zero of solutions of an ODE naturally associated to the problem. This theorem implies some classical oscillation criteria of W. Ambrose and R. Moore. Extension to constant higher-order mean curvature hypersurfaces are also presented.