1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

286 35. STABILITY OF RICCI FLOW

of the associated Lie algebra g. (See Part I , Chapter 1, §6.) Every algebraic soliton

on a simply-connected homogeneous space is an expanding Ricci soliton. (See

[177] or [182].) However, to analyze the linear stability of expanding Ricci solitons,

understood as stationary solutions of dilated Ricci fl.ow, it is more effective to adopt

this more algebraic perspective. By using this formulation, Jablonski, Petersen,

and Williams have recently proved linear stability for a large class of expanding

homogeneous solitons [157], greatly extending earlier work of three of the authors

of this work [129]. Jablonski, Petersen, and Williams prove, for example, that all

nilsolitons^4 of dimensions n ::::; 6 as well as some families of solvsolitons are strictly

linearly stable; we refer the reader to [157] for precise statements.

1. 
   
   
   
   7. Linear stability of Perelman's average energy.
      Cao, Hamilton, and Ilmanen [47] have proved a result that illustrates the in-
      teresting fact that the linear stability of Perelman's average energy functional is
      equivalent to the linear stability of the Ricci fl.ow itself.
   
   
   
   

If (Mn, g) is a compact manifold and f : Mn ---+ JR is a smooth function,

Perelman [312] defines the average energy functional

F(g, f) = { (l\7f1^2 + R)e-f dμ

}Mn

and the diffeomorphism invariant quantity

(35.8)

As shown in Chapter 5 of Part I, the gradient fl.ow of F is equivalent to the coupled

modified Ricci fl.ow

8

8tg = -2(Rc+\72 f),

(%t + ~ + R) f = 0.

Conversely, this fl.ow is equivalent (up to diffeomorphism) to the system

8

-g 8t = -2Rc ,

( :t + ~ + R) f = l\7 fl

2

·

Furthermore, the critical points of the gradient fl.ow are precisely the compact
steady gradient solitons. All such metrics are necessarily Einstein, that is, Ricci
fl.at. Because of this correspondence, it is not surprising that the second variation
of ,\ at a critical point is intimately related to the first variation of Ricci fl.ow at a
fixed point.

PROPOSITION 35.13 ([47]). Let (Mn, g) be a compact Ricci-fiat manifold. Then

the second variation of Perelman's .-\-invariant is given by

D^2 .-(g) (h , h) = j (Lh, h) dμ,

(^4) A soliton on a nilpotent group is called a nilsoliton, and a soliton on a solvable group is
called a solvsoliton.