A maximum principle at infinity with applications to geometric vector fields

By L. Alías, A. Caminha, F. do Nascimento. Journal of Mathematical Analysis and Applications, 2019

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Abstract We derive a new form of maximum principle, applicable to a vector field with nonnegative divergence in a connected, oriented, complete noncompact Riemannian manifold. We then use it to obtain some applications to Killing vector fields. More precisely, we first show that, under a reasonable condition at infinity, an orientable, connected, complete noncompact hypersurface of a Riemannian manifold, transversal to a Killing vector field of constant norm and with nonnegative second fundamental form, is totally geodesic. We also deal with the case of a hypersurface of constant mean curvature – instead of nonnegative second fundamental form, and show that it has to be totally geodesic too.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/543d38ba1f27ed0b9a2eb5978d386dcd7b424334