Remarks on non-compact gradient Ricci solitons

2009/05/18 by Stefano Pigola, Pigola, Stefano, Michele Rimoldi +3 · 12 citations
Mathematics · #53C21 #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.0905.2868

openalex publication_date 2009/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L^p-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under L^p conditions on the relevant quantities.

Citations

• Ricci solitons: the equation point of view
• Noncompact Shrinking 4-Solitons with Nonnegative Curvature

Cited by

• Rigidity of weighted manifolds via classification results for semilinear equations
• Curvature estimates for steady and expanding solitons in higher dimensions
• Linear stability and instability of Kähler Ricci solitons
• Rigidity of Gradient Shrinking Ricci Solitons with a Vanishing Bach-like Tensor and Related Variational Formulas
• On curvature estimates for four-dimensional gradient Ricci solitons
• Curvature pinching of asymptotically conical gradient expanding Ricci solitons
• Gradient Shrinking Ricci Solitons and Modified Sectional Curvature
• Geometric Structure of Ends of Ricci Shrinkers
• Hamilton's identity and rigidity of complete gradient solitons
• Rigidity of Five-Dimensional shrinking gradient Ricci solitons
• An eigenvalue estimate for self-shrinkers in a Ricci shirinker
• Four-dimensional shrinkers with nonnegative Ricci curvature

Related

• Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature
• A criterion for the triviality of non-steady gradient Ricci solitons
• On Ricci solitons whose potential is convex
• Rigidity of gradient Ricci Solitons
• On gradient Ricci solitons