- ANALYTIC SEMIGROUPS AND MAXIMAL REGULARITY THEORY 295
In this case, a standard Schauder estimate applied to the operator )...J - A (g) yields
C < oo such that
llrJll£ 1 ~ llrJllh^2 +u ~ C II ( )..J - A (g)) r]llho+u ~ C II ( >J - A (g)) r/11£o
for every rJ E E 1 = D(A (g)). Using Theorem 1.2.2 and R emark 1.2.l(a) of [6], we
see that the result follows from this estimate. O2.3. A center manifold theorem.
If A is a well-behaved quasilinear differentia l operator acting on appropriate
function spaces and if its linearization DA(u) at a fixed point u h as an eigenvalue
on the imaginary axis , then one exp ects the evolution of solutions starting near
that fixed point to b e characterized by the presence of exponentially attractive (or
repulsive) center m anifolds. Below , we state the Center Manifold Theorem, which
makes this intuition precise and is an adaptation of results^5 in [383]. However the
statement of the theorem is rather technical, and so we fir st give a rough description
of its ingredients and conclusions.
The theorem follows from maximal regularity theory, and we therefore need
to define appropriate interpolation spaces; its application a lso requires that the
extension of the operator generate an a n alytic semigroup. We must work loca lly to
apply the implicit function theorem, and to this end we use the space Q°'. This is
the content of the hypothes is of the theorem.
The t a ngent space to the space of metrics at the fixed point g must be de-
composable into stable and (center) unstable subspaces, b ased on the signs of the
real p arts of the eigenvalues of the operator. We expect unstable behavior on the
subspace of eigenvectors associated with positive eigenvalues, stable b ehavior on
the subsp ace associated with negative eigenvalues, and center manifold b eh avior
(possibly stable or unstable) if the eigenvalue h as zero real part. The content of
conclusion (1) of Theorem 35.27 b elow is that one can indeed decompose the space
into these parts.
Conclusion (2) of Theorem 35.27 says that there exist s a (local) center m a nifold
with locally invariant behavior. That is, there exists a small neighborhood such that
if the solution is in that n eighborhood, it can only move a long the center ma nifold.
For Ricci flow we can make an additional argument that once solutions land there
they stay there, but t h at is not the case in general.
From the variation of constants formula obtained via maximal regularity theory,
we expect to obtain estimates of the solution in terms of its initial value, as well as
its exponentia l b eh avior; this is the content of conclusion (3).
We are ready for the statement of Simonett's center manifold theorem (note
that B (X, x , d) denotes the open b all of radius d centered at x in a metric space
X). For simplicity, we present a version of the theorem that does not include the
optimal boost in regularity.
THEOREM 35.27 (Center Manifold Theorem). Let X 1 '---+ Xo be a continuous
dense inclusion of Banach spaces, and let Xa and X/3 denote the continuous inter-polation spaces corresponding to fi xed 0 < f3 < a < 1. Let
8
(35.19) otg =A (g) g(^5) See, in particular, Theorems 4. 1 and 5.8 and Remark 4.2 in [383].