- LINEAR STABILITY OF RICCI FLOW 283
If g = g for a fixed point g of Ricci flow, then the Lichnerowicz Laplacian 6.L
is the natural linearization of Ricci- DeTurck flow:PROPOSITION 35.3. The choice g = g yields
(35. 7)
where
(6.Lh)ij ~ 6.hij + 2Rkijehke - Rikhkj - R1khki·The operator 6.L is self-adjoint and strictly elliptic. Furthermore, if Mn is com-
pact, then it is self-adjoint with respect to the L^2 inner product (-, ·) ~ J Mn (-, ·) dμ.
The only stationary solutions of Ricci flow are Ricci-flat metrics, and the only
stationary solutions of normalized Ricci flow are Einstein metrics. However, Ricci
flow has a wealth of "generalized fixed points" , namely Ricci solitons. It is often
useful to study the stability of these solut ions using a modified flow.
Shrinking solitons are converted into stationary solutions as follows. (We use
this calculation below in Subsections 1.6 and 3.2.) If (Mn, g(t)) is any solution ofRicci flow that becomes singular at some T < oo and if X is any smooth vector
field on Mn, we may introduce a dilated time variable T := - 2 \ log(T-t) and a
rescaled family of metrics 1( T) defined by
g(t) := 2>-.(T-t)(<p;/(T(t))),where 'Pt denotes the 1-parameter family of diffeomorphisms generated by the vectorfields (2>..(T-t))-^1 X, and>..> 0. Then one calculates
:tg(t) = :t (2>..(T - t)¢;l(T(t)))
= -2>..¢;(1(T(t)) + 2>..(T- t)) [¢; (~~ ~:) + : 8 ls=o(¢;+sh(T(t))))]
= ¢; (-2>..1(T(t)) + ~~ + .Cx/(T(t))).
Because
8
atg = -2 Rc(g) = ¢; (-2 Re(!)),we see that / evolves by dilated Ricci flow,
8
07
/ = -2Rc[IJ - .Cx1 + 2>..1.Clearly, any suitably normalized shrinking soliton (Mn,/, X) becomes a stationary
solution of the dilated Ricci flow constructed in this manner.
EXERCISE 35.4. Given a solution (Mn,g(t)) of Ricci flow that emerges from
a "big bang" at some time T 0 > -oo, and a smooth vector field X, show that
replacing T - t by t -To in the construction above leads to a solution g(T) of
:
7g = - 2Rc[g] - .Cx[J - 2A.g,
where here T := 2 \ log(t - T 0 ). In this way, any suitably normalized expanding
soliton becomes a stationary solution of a modified flow. HINT: See §3.1 of [129].