296 35. STABILITY OF RICCI FLOW
be an autonomous quasilinear parabolic equation posed for t ?': 0 such that A ( ·) E
Ck (Q f3, [, ( X 1 , X 0 )) for some positive integer k and some open subset Q f3 s;;; X13.
Assume that there exists a pair E:o 2 £ 1 of Banach spaces and extensions A ( ·) of
A(-) to domains D(A (-)) that are dense in £ 0 such that the following conditions
hold for each g E Ycx ~ Q13 n Xcx:
- A (g) E [, ( £ 1 , £ 0 ) generates a strongly continuous analytic semigroup on
[, (E:o); - Xo ~ (£ 0 , D(A (g)))e and X 1 ~ (£ 0 , D(A (g)))(l+o) for some BE (0 , l);
- A (g) agrees with the restriction of A (g) to the dense subset D (A) s;;; Xo;
- £ 1 Y X13 Y £ 0 is a continuous and dense inclusion with the property that
there are C > 0 and o E (0, 1) such that for all TJ E £1, one has
llTJllx" :::; C llTJll~~
8
llTJll~ 1 ·Let g E Ycx be a fixed point of (35.19). Suppose that the spectrum~ of the linearized
operator D 9 A admits the decomposition~ = ~sU~c, where ~s C { z : Re z < O} and
where ~c C { z : Re z ?': 0} consists of finitely many eigenvalues of finite multiplicity.
Suppose further that ~c n i~-/= 0. Then the following hold:
(1) If S (>.) denotes the eigenspace corresponding to A E ~c, then Xcx ad-mits the decomposition Xcx = x; EB xi for all a E [O, l], where xi =
EB .\EEc S (A)·(2) For each r E N, there exists dr > 0 such that for all d E (0, dr], there
is a bounded er map ~[r,dJ : B (Xf, g , d) ---+ X{ with ~[r,dJ (g) = 0 and
D 9 ~ = 0. The image of ~[r,d] lies in the closed ball B ( X{, g , d), and its
graph is a er manifold
Mf 0 c ~ { (/', ~[r,dJ (1)) : r EB (X[, g, d)} c X1
satisfying T 9 M\ 0 c ~ Xf. One says Mfoc is a local center manifold
if ~c C i~ and a local center unstable manifold otherwise. Mf 0 c
is locally invariant for solutions of (35.19) as long as they remain in
B(Xf, g ,d) x B(X{,O,d).(3) For all a E (0, 1), there are constants Ccx > 0 independent of g and
constants w > 0 and d E (0, do] such that for each d E (0, d], one has
ll7r^5 g (t) - ~[r,d] (7rcg (t))llx, :::; /!__cxcx e-wt ll7r^5 g (0) - ~[r,d] (7rcg (O))llxafor all solutions g(t) with g(O) E B(Xcx,g,d) and all times t ?': 0 such
that the solution g (t) remains in B (Xcx, g , d). Here 7r^8 and 7rc denote theprojections onto x; ~ ( X{, X 0 ) ex ~nd xi, respectively.
The reader is strongly cautioned that local center (unstable) manifolds are
not in general unique. For instance, it may be the case that dr ---7 0 as r ---7 oo.
Furthermore, the Center Manifold Theorem does not in general tell us anything
about the dynamics within a local center (unstable) manifold. In particular, it is
not true in general that center manifolds consist of stationary solutions.
3. Dynamic st ability results obtained us ing linearization
3 .1. Stability of Ricci fl.ow at a fl.at met r ic.
We are now ready to illustrate the method described above by establishing the
dynamic stability of Ricci fl.ow at a fiat compact manifold (Mn, g). We shall find