Complete gradient shrinking Ricci solitons have finite topological type

By F. Fang, Jessica Man, Zhenlei Zhang. 2007

50 citations recorded by the source corpus.

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-\'{E}mery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero. Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/92836e14e8fffc1f65b1608ee3bde2af926ecc3f

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

On Gradient Ricci Solitons (2009, background): “We note that asymptotic estimates for the potential function f were previously studied in [27, 25, 26, 12] and interesting volume growth properties of solitons were also investigated in [8].”
Geometry of Complete Gradient Shrinking Ricci Solitons (2009, background): “Moreover, Fang-Man-Zhang [18] proved that a complete gradient shrinking Ricci soliton with bounded scalar curvature has finite topological type.”
Sharp logarithmic Sobolev inequalities on gradient solitons and applications (2008, background): “This follows from the observation that f is a proper function and has no critical point outside a compact subset [17], since for any x ∈ M and γ(s) a minimizing geodesic jointing o ∈ M , a fixed point, to x, with f(γ(0)) = o and f(γ(s0)) =x”