Smooth metric measure spaces with non-negative curvature

2011/03/03 by Ovidiu Munteanu, Jiaping Wang, Munteanu, Ovidiu +1 · 8 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1103.0746

openalex publication_date 2011/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study both function theoretic and spectral properties on complete noncompact smooth metric measure space (M,g,e^-fdv) with nonnegative Bakry-Émery Ricci curvature. Among other things, we derive a gradient estimate for positive f-harmonic functions and obtain as a consequence the strong Liouville property under the optimal sublinear growth assumption on f. We also establish a sharp upper bound of the bottom spectrum of the f-Laplacian in terms of the linear growth rate of f. Moreover, we show that if equality holds and M is not connected at infinity, then M must be a cylinder. As an application, we conclude steady Ricci solitons must be connected at infinity.

Citations

• Some Geometry and Analysis on Ricci Solitons
• Differential equations on riemannian manifolds and their geometric applications

Cited by

• Rigidity of weighted manifolds via classification results for semilinear equations
• Gradient estimates and Liouville properties for the drifted Laplacian
• Curvature estimates for steady and expanding solitons in higher dimensions
• Gradient estimates for (p,V)-harmonic functions on Riemannian manifolds
• Remark on a geometric inequality for closed hypersurfaces in weighted manifolds
• Cohomogeneity One Expanding Ricci Solitons and the Expander Degree
• Topology at infinity of complete gradient Schouten solitons
• Geometric Structure of Ends of Ricci Shrinkers

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