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Gaussian densities and stability for some Ricci solitons
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abstract
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.
Forward citations
Cited by 2 Pith papers
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Open full-metric formation and local marked first-order asymptotic moduli of FIK blowdown singularities
FIK blow-down singularities form from an open family of nearby Ricci flow initial data on any closed four-manifold, and nearby flows carry a local first-order asymptotic coordinate.
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Linear stability of Perelman's $\nu$-entropy of standard Einstein manifolds
The authors prove λ1 > 2E for almost all non-symmetric standard Einstein manifolds G/H with G simple, implying Schwahn's stable examples are linearly stable for Perelman's ν-entropy.
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