Some Geometry and Analysis on Ricci Solitons

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.

8 pages

References in corpus (2)

• Ricci flow with surgery on three-manifolds
• Ricci Flow and the Poincare Conjecture

Cited by in corpus (8)

• Smooth metric measure spaces with non-negative curvature
• Comparison Geometry for the Bakry-Emery Ricci Tensor
• Rigidity of gradient Ricci Solitons
• Isometry theorem of gradient Shrinking Ricci solitons
• On shrinking Gradient Ricci Soliton With Nonnegative Sectional Curvature
• Complete Shrinking Ricci Solitons have Finite Fundamental Group
• A note on lower bounds for the first eigenvalue of the Witten-Laplacian
• Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors