Some Geometry and Analysis on Ricci Solitons

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.