1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. DYNAMIC STABILITY RESULTS OBTAINED USING LINEARIZATION 297

that g b elongs to a unique exponentially attractive C^00 center manifold consist-

ing exactly of flat metrics. Furthermore, any solution starting sufficiently near g

converges to a flat metric in this center manifold.

We begin with the observation that stability of the Ricci flow follows from the

stability of the Ricci- DeTurck flow with background metric g = g:

(35.20a)

(35.20b)

8

otg = A 9 g = -2Rc (g) - P 9 (g),

g (0) =go.

LEMMA 35.28. Let W (t) be a vector field on a compact Riemannian manifold

(Mn, g (t)), where 0 :St< oo, and suppose there exist 0 < c :SC< oo such that

sup IW (x, t)lg(t) :S c e - ct.

xEM

Then the diffeomorphisms 'Pt generated by W converge exponentially to a fixed

diffeomorphism cp 00 of Mn.

PROOF. Given x E M, let 1 : [O, oo) ---+ Mn be an integral curve for W starting

at x. Then 1 satisfies

1

1

( t) = w h ( t) ' t) '

1 (0) = x,

where we make the standard identification 1' = 1• (d/dt). The length Lt bl of the

integral curve is nondecreasing and bounded a bove b ecause

1

t 1t c c

L t bl= IW (x, T)lg(r) dT :SC e-cr dT = - (1 - e-ct) :S -.

0 0 c c

This proves that L t bl converges as t ---+ oo. To see t hat the convergence is expo-

nential, it suffices to note that

1

00 1

00

IW (x, T)lg(r) dT :SC e - cr dT = - e- ct. c

t t c

Since 1 (t) ='Pt (x) and since x E Mn is arbitrary, the result follows. 0

COROLLARY 35.29. Let g be a fiat metric on a compact manifold Mn. Suppose

there is a neighborhood 0 of g in si with respect to the ll·lb+P Holder norm such

that for every g 0 E 0 , the unique solution g (t) of the Ricci- D e Turck flow (35.20)

converges exponentially fast to a fiat metric g 00 • Then the unique solution g (t) ~

cp; g of the Ricci flow with the same initial data g (0) =go E 0 converges exponen-

tially fast to a fiat metric ?Joo.

PROOF. It is clear that § 00 must b e flat if it exists, so all we need to do is show

that g (t) converges. But because g 00 and g are both flat, one has

V'[goo]g = Y'[g]g = 0.

Then since g (t) ---+ g 00 exponentially fast, it follows that the infinitesimal generators
W (t) of the DeTurck diffeomorphisms vanish exponentially, where

W i ....:... ...,... g i j (--1g ) jk g g kf pq (n v p9qe -- 2 1 n -v egpq ).

(Here V' denotes covariant differentiation with respect to the Levi-Civita connec-
t ion of g (t).) Hence it follows from the lemma that the solution g (t) converges
exponentially fast to some limit § 00. 0