Gaussian densities and stability for some Ricci solitons

Details Gaussian densities and stability for some Ricci solitons Gaussian densities and stability for some Ricci solitons

2004-04-07

arxiv.org/abs/math/0404165v1

Mathematics

Differential Geometry

15 pages

Scientific paper

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.

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