Some Geometry and Analysis on Ricci Solitons

2006/12/18 by Aaron Naber, Naber, Aaron · 3 citations
Mathematics · Physics and Astronomy · #53C21 #53C44 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.math/0612532

openalex publication_date 2006/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Bakry-Emery Ricci tensor of a metric-measure space (M,g,e^-fdv_g) plays an important role in both geometric measure theory and the study of Hamilton's Ricci flow. Under a uniform positivity condition on this tensor and with bounded Ricci curvature we show the underlying space has finite f-volume. As a consequence such manifolds, including shrinking Ricci solitons, have finite fundamental group. The analysis can be extended to classify shrinking solitons under convexity or concavity assumptions on the measure function.

Citations

• Ricci flow with surgery on three-manifolds

Cited by

• On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions
• Smooth metric measure spaces with non-negative curvature
• Complete Shrinking Ricci Solitons have Finite Fundamental Group

Related

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• Complete gradient shrinking Ricci solitons have finite topological type
• The Soliton-Ricci Flow over Compact Manifolds
• Analysis of weighted Laplacian and applications to Ricci solitons
• On compact Ricci solitons in Finsler geometry