Some Geometry and Analysis on Ricci Solitons

Details Some Geometry and Analysis on Ricci Solitons Some Geometry and Analysis on Ricci Solitons

2006-12-18

arxiv.org/abs/math/0612532v1

Mathematics

Differential Geometry

8 pages

Scientific paper

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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