Weighted volume growth and vanishing properties of f-minimal hypersurfaces in a weighted manifold. (March 2019) - Yun, Gabjin; Seo, Keomkyo
Abstract: In this paper, we prove that a complete noncompact submanifold in a weighted manifold with nonpositive sectional curvature has at least linear weighted volume growth. Moreover we obtain several sufficient conditions for f -minimal hypersurfaces to have infinite weighted volume. By using an f -Laplacian comparison result, we obtain a lower bound of the first eigenvalue for the f -Laplace operator on submanifolds in a weighted manifold. We also obtain vanishing results for L f 2 harmonic 1-forms on complete noncompact f -minimal hypersurfaces in a weighted manifold. Finally we prove a topological structure theorem for complete noncompact L f -stable f -minimal hypersurfaces via a Liouville-type theorem for f -harmonic functions with finite f -energy.