Weighted volume growth and vanishing properties of f-minimal hypersurfaces in a weighted manifold. (March 2019)

• Record Type: Journal Article
• Authors: Yun, Gabjin; Seo, Keomkyo
• Is Part Of: Nonlinear analysis. Volume 180(2019)
• Page Start: 264
• Page End: 283
• Publication Date: 2019-03
• Subjects: 53C42 -- 53C21 -- 58C40; f-minimal hypersurface -- Weighted manifold -- Stability -- f-Laplacian -- First eigenvalue -- Harmonic form; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.10.015
• Languages: English
• ISSNs: 0362-546X

Abstract

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.