Linear stability of Perelman's ν-entropy on symmetric spaces of compact type

Abstract

Following [`Gaussian densities and stability for some Ricci solitons', preprint 2004], in this paper we study the linear stability of Perelman's ν-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability (also called ν-stability in this paper) restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the ν-entropy on symmetric spaces of compact type. In particular, we exhibit many more ν-stable and ν-unstable examples than previously known and also the first ν-stable examples, other than the standard spheres, whose second variations are negative definite.

On the stability of Einstein manifolds

“…In [AM11], Andersson and Moncrief prove that the Lorentzian cone over a compact negative Einstein metric is an attractor of the Einstein flow under the assumption that the compact Einstein metric is stable. Stability also appears in the context of the Ricci flow and its analysis close to Einstein metrics [Ye93,CHI04,Ses06,Has12,CH13]. This is because the second variational formulas of Perelman's entropies on Einstein metrics are closely related to the second variational formula of the Einstein Hilbert action.…”

“…Most symmetric spaces of compact type (including the sphere and the complex projective space) are stable [Koi80,CH13]. Spin manifolds admitting a nonzero parallel spinor are stable [Wan91,DWW05].…”

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

“…There has been a good deal of work on determining the stability of Einstein metrics and it is expected that stable geometries should be quite special. By combining the above result with the work of Cao and He , we now have a complete understanding of Ricci flow stability for the canonical Einstein metrics on the compact, connected, simply connected, simple Lie groups. Corollary If is a compact, connected, simply connected, simple Lie group and the bi‐invariant metric is dynamically stable then is one of: , for , , , or . If the metric is unstable, then is one of: for , for , . …”

“…The stability of all of the groups in Corollary except can be determined by studying the variational stability of Perelman's functional; this is what was achieved by Cao and He in . The reason that the stability of is not accessible by such analysis is that it admits certain neutral directions of deformation coming from conformal variations of the metric.…”

Stability of compact Ricci solitons under Ricci flow

“…Since Ricci solitons are critical points of Perelman's λ-entropy and ν -entropy, they displayed the second variation of λ and ν functionals and, according to the second variation, explored the linear stability of some examples. For more results on stability of Ricci solitons with respect to the second variation of Perelman's ν -functional see [6,5,13].…”