Some Geometry and Analysis on Ricci Solitons
Aaron Naber

TL;DR
This paper investigates the properties of Ricci solitons and metric-measure spaces with positive Bakry-Emery Ricci tensor, establishing finiteness results for volume and fundamental group, and classifying shrinking solitons under convexity or concavity conditions.
Contribution
It provides new results on the finiteness of volume and fundamental group for spaces with positive Bakry-Emery Ricci tensor and classifies shrinking Ricci solitons under specific measure function conditions.
Findings
Spaces with positive Bakry-Emery Ricci tensor have finite f-volume.
Such manifolds, including shrinking Ricci solitons, have finite fundamental group.
Classification of shrinking solitons under convexity or concavity assumptions on the measure function.
Abstract
The Bakry-Emery Ricci tensor of a metric-measure space (M,g,e^{-f}dv_{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.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Advanced Differential Geometry Research
