Gaussian densities and stability for some Ricci solitons
By Huaihuo Cao, R. Hamilton, T. Ilmanen. 2004
138 citations recorded by the source corpus.
In this announcement, we exhibit the second variation of Perelman's $\lambda$ and $\nu$ functionals for the Ricci flow, and investigate the linear stability of examples. We also define the "central density" of a shrinking Ricci soliton and compute its values for certain examples in dimension 4. Using these tools, one can sometimes predict or limit the formation of singularities in the Ricci flow. In particular, we show that certain Einstein manifolds are unstable for the Ricci flow in the sense that generic perturbations acquire higher entropy and thus can never return near the original metric.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/1f40bb1acd1436da960e91cefa5c9b73ea70b10b
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In Applied Mathematics
In Astronomy and Astrophysics
In Geometry and Topology
Citations in the corpus
- On second variation of Perelman’s Ricci shrinker entropy (2010, background): “In [4], Hamilton, Ilmanen and the first author initiated the study of linear stability of Ricci solitons.”
- Recent Progress on Ricci Solitons (2009, background): “ature has finite topological type. 6. Stability of Ricci Solitons In this section we describe the second variation formulas for Perelman’s λ-energy and ν-energy due to Hamilton, Ilmanen and the author [16]. 6.1. Second Variation of λ-energy. Recall that the λ-energy is defined by λ(gij) = inf{F(gij,f)”
- On type-I singularities in Ricci flow (2010, background): “Finally, in the last section we define a density function θp,T for Type I Ricci flows, related to the central density for gradient shrinking solitons defined by Cao-Hamilton-Ilmanen [4], and prove a regularity type theorem (Theorem 4.”