298 35. STABILITY OF RICCI FLOW
Combining this corollary together with Theorem 35.27, we obtain the following
stability result for fl.at metrics under Ricci fl.ow:
THEOREM 35.30 (Asymptotic stability at Rm = 0). Let (Mn,g) be a fiat
compact Riemannian manifold. For fi xed p E (0, 1), let X denote the closure of52 => 5t with respect to the ll·ll 2 +P Holder norm. Then:
(1) T 9 5t ~ X admits the decomposition
rr:5l g (^2) + - ,.ys "" w ,ye )
where x e is the n (n + 1) /2-dimensional space of (2, 0)-tensors parallel
with respect to the Levi-Civita connection of g.
(2) There exists do > 0 such that for all d E (0, d 0 ], th er e is a bounded C^00
map
'ljJ : B (,ye, g, d) --+ x s
such that 'ljJ (g) = 0 and D 9 '1jJ = 0. The image of 'ljJ lies in the closed ball
13 (X^8 , g, d), and its graphMfoe ~fry, 'l/J (r)): IE B (Xe, g, d)}
satisfies T 9 Mf 0 e ~ x e. In other words, Mfoe is a unique C^00 local center
manifold of dimension n (n + 1) / 2 that consists entire ly of fiat metrics.
(3) There are constants C > 0, w > 0, and d* E (0, d 0 ] such that for each
d E (0, d*], one has
llnsg (t) - 'l/J (neg (t))ll 2 +P ~Ce-wt llnsg (0) - 'l/J (neg (O))ll 2 +P
for all so lutions g (t) of the Riccifiow with g (0) EB (X, g, d) and all times
t ~ 0. Here n^8 and 7re denote the projections onto xs and xe, respectively.In particular, any solution g (t) of Ricci fiow with initial data sufficiently
near g converges exponentially to a fiat metric near g.PROOF. By passing to a finite cover, we may assume that Mn is a torus. We
shall apply the Center Manifold Theorem to t he Ricci- DeTurck flow (35.20) and
then use Corollary 35. 29 to form a conclusion about the Ricci flow itself.
Recall that any fl.at metric is a stationary solution of the Ricci-DeTurck flow.
In the previous section, we proved that DA 9 (g) is the rough Laplacian. This
operator is negative semidefinite on 52 with its kernel consisting exactly of the
parallel (2, 0)-tenso rs, hence of dimension at most n (n + 1) /2. For the choices we
have made above for Xo, X 1 , Ea, and £ 1 (i.e., X 0 = h^0 +P,X 1 = h^2 +P, £ 0 = ho+u,
and £ 1 = h^2 +u, for a fixed choice of u and p satisfying 0 < u < p < 1, with the
little Holder spaces defined in Definition 35.21) the results of Lemmas 35.25 and
35.26 allow us to apply the Center Manifold Theorem t o the operator A 9. It follows
that local er center manifolds r Mfoe exist and that the Ricci- DeTurck flow of any
metric starting near g exponentially approaches r Mfoc·
We now claim that the spaces r Mfoc are independent of r and consist entirely
of flat metrics. To prove this, we observe that any flat metric g sufficiently near g
belongs to r Mfoe for all r E N: If not, then statement (3) of the Center Manifold
Theorem would imply that g converges exponentially to r Mfoe> contradicting the
fact that g is a stationary solution. The space of flat metrics on the torus is a convex
n (n + 1) /2-dimensional set. Since each r Mf 0 e is at most n (n + 1) / 2-dimensional,
it follows that r Mf 0 c co nsists exactly of the fl.at metrics for all r E N.