Complete Shrinking Ricci Solitons have Finite Fundamental Group
By W. Wylie. 2007
110 citations recorded by the source corpus.
We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of Garcia-Rio and Fernandez-Lopez in the compact case.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/408635b11229e114d7c9336444b5536856ced6e6
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In Applied Mathematics
In Astronomy and Astrophysics
In Geometry and Topology
Citations in the corpus
- On Gradient Ricci Solitons (2009, result): “Some other results about the topology of shrinking Ricci solitons have been obtained in [30, 12].”
- Geometry of Complete Gradient Shrinking Ricci Solitons (2009, background): “On the topological side, Wylie [41] showed that a complete shrinking Ricci soliton has finite fundamental group.”
- Ricci almost solitons (2010, background): “Following [32], let us consider the Riemannian universal covering P : M ′ → M of M .”
- Complete gradient shrinking Ricci solitons have finite topological type (2007, background): “Suppose d(p,q) = L ≥ �. Let '(t) : [0,L] → [0,1] be an arcwise smooth function such that '(0) = '(L) = 0. By the second variation formula, as did in [9] or [11], we have the following estimate:”
- Recent Progress on Ricci Solitons (2009, background): “Wylie [97] showed that a complete shrinking Ricci soliton has finite fundamental group.”