Weighted volume growth and vanishing properties of f-minimal hypersurfaces in a weighted manifold

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. (C) 2018 Elsevier Ltd. All rights reserved.