1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. ANALYTIC SEMIGROUPS AND MAXIMAL REGULARITY THEORY 287

where

h EH,

h EV.

In [47], Cao, Hamilton, and Ilmanen also study the second variation of the

infimum v of Perelman's entropy functional W. They show in particular that ClPn

is linearly stable (but not strictly so) with respect to v , but that any positively

curved Kahler-Einstein manifold (Mn, g) with dim H(l,l) (M) 2: 2 is unstable. Hall

and Murphy [133] extend this result to shrinking Kahler- Ricci solitons, showing

that any such manifold with dim H(l,l) (M) 2: 2 is also unstable with respect to

v. A corollary of t he Hall-Murphy result is that the Koiso- Cao soliton [168], [44]

2 - 2 2 -2

on CJP # CJP and the Wang- Zhu soliton [431] on CJP # (2CJP ) are both linearly

unstable.

1.8. Dynamic stability.

Fix a compact manifold Mn and let 9J1 denote a normed space of Riemannian

metrics on Mn. Suppose g E 9J1 is a fixed point of a geometric evolution equation

(35.9) { gtg = F(g),

g(O) =go.

In t he rest of this chapter, we are interested in dynamic stability at g, which we

define in two forms:

DEFINITION 35.14. A stationary solution g of (35.9) is

• asymptotically stable if there is an open set U ~ 9J1 containing g such

that every solution of (35.9) with g 0 E U exists for all positive time and

converges to g as t -7 oo,

• stable if for every open set V ~ 9J1 containing g, there exists an open set

U ~ V containing g such that every solution of (35.9) with g 0 E U exists

for all positive time and stays in V.

We emphasize that passing from linear stability to one of these two forms of
dynamic stability can be highly nontrivial. In §2, we present an approach that
accomplishes this, particularly in the presence of a center manifold. In §3, we
survey some successful applications of this and similar theories. Finally, in §4, we
survey various dynamic stability results obtained via alternate approaches.

2. Analytic semigroups and maximal regularity theory
   Tools to pass from linear stability to dynamic stability using semigroups were
   first developed for semilinear autonomous PDE (see, for example, [149] and [81]).
   These techniques have been extended to fully nonlinear autonomous equations in
   the work of Da Prato and Lunardi [92]. The case in which the linearization pos-
   sesses a null eigenvalue is studied for qu asilinear diffusion-reaction equations in
   Simonett [383]. Simonett's method also provides t he "regularity boost" noted in
   the introduction.
   To move from linear stability to dynamic stability, each of the nonlinear meth-
   ods essentially involves an application of the implicit function theorem, in some