Gaussian densities and stability for some Ricci solitons

Perelman [Pe02] has discovered a remarkable variational structure for the Ricci flow: it can be viewed as the gradient flow of the entropy functional λ. There are also two monotonicity formulas of shrinking or localizing type: the shrinking entropy ν, and the reduced volume. Either of these can be seen as the analogue of Huisken’s monotonicity formula for mean curvature flow [Hu90]. In various settings, they can be used to show that centered rescalings converge subsequentially to shrinking solitons, which function as idealized models for singularity formation. In this note, we exhibit the second variation of the λ and ν functionals, and investigate the linear stability of examples. We also define the “central density” of a shrinking Ricci soliton (shrinker) and compute its value 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. A detailed version of the calculations summarized in this announcement will follow in [CHI]. In §1, we investigate the stability of Perelman’s λ-functional. Its critical points are steady solitons (Ricci flat in the compact case). We compute the second variation D2λ; the corresponding Jacobi field operator L is a degenerate negative elliptic integro-differential operator. In fact, L equals half the Lichnerowicz Laplacian ∆L on divergence-free symmetric tensors, and zero on Lie derivatives. This fact and further investigations of the second variation have been reported by Perelman [Pe03]. We call a steady soliton linearly stable if L ≤ 0, otherwise linearly unstable. If g is linearly unstable, then g can be perturbed so that λ(g) > 0, which will destabilize it utterly: it will decay into a