Stability and instability of Ricci solitons

Abstract

We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton (M, g) is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists for all time and converges towards a Ricci soliton. If g is not a local maximum of the shrinker entropy, we show that there exists a nontrivial normalized Ricci flow emerging from it. These theorems are analogues of results in the Ricci-flat and in the Einstein case [HM13,Krö13b].

Stability of Riemannian manifolds with Killing spinors

“…Moreover, S-linear instability implies ν-linear instability and further also ν-instability (see Definition 1.2 below). Then by Theorem 1.3 in [Kr15], (since Λ > 0) it also implies that g is dynamically unstable for the Ricci flow.…”

The canonical Einstein metric on G2 is dynamically unstable under the Ricci flow

“…Hence, in the course of the proof of Theorem , we show that has non‐integrable infinitesimal solitonic deformations. This can be compared with [, Theorem 1.5] where it is shown that the Fubini–Study metric on has non‐integrable solitonic deformations for .…”

“…The relationship between Cao–Hamilton–Ilmanen's linear stability and dynamical stability for Einstein metrics with was made precise by Kröncke who built on earlier work by Sesum and Haslhofer and Müller . Proposition Let be a compact Einstein manifold satisfying with .…”

“…Remark In , the results are stated with a further condition that has to satisfy, namely that the infinitesimal solitonic deformations must be integrable (that is, the deformation comes from a genuine curve of Ricci solitons). As our notion of linear stability from Proposition does not include the possibility that the Lichnerowicz Laplacian restricted to has an eigenvalue equal to , has no such deformations and there is nothing to check, so we omit this condition from the statement of Proposition .…”

Stability of Einstein Metrics on Fiber Bundles

“…On the other hand, linear instability along a nonzero TT-tensor gives a non-trivial variation along which S has a local minimum. When E > 0, Theorem 1.3 in [Kro15] then implies that such an Einstein metric is dynamically unstable. Therefore, all linearly unstable Einstein metrics discussed in this paper are dynamically unstable, for example Einstein metrics investigated in Corollaries 1.3 and 1.4.…”