280 35. STABILITY OF RICCI FLOW
one expects to encounter a center manifold on which the behavior of solutions is
determined not by the linearization, but by higher-order terms in A.
In a broad sense, any convergence theorem that states that any solution evolved
from initial data in some open set converges to a unique fixed point in that set, such
as the results proved for the Ricci flow by Hamilton [135], [136], Huisken [152],
Margerin [217], [218], Nishikawa [294], Ye [449], Bohm and Wilking [30], Chen
[65], and Brendle and Schoen [35] (we survey all of these in §4 of this chapter),
implies a statement of stability. In this chapter however, we are primarily concerned
with techniques for determining stability that start with the known existence of a
stationary solution.
This chapter then serves two purposes. The first is to give a detailed look at
one approach to stability, namely that of linearizing the equation and then applying
semigroup and maximal regularity theory to determine dynamical stability. The
second purpose is to survey related stability results that are obtained using different
methods. The level of detail varies correspondingly.
In the first (main) part of this chapter- §1 and §2- we focus on methods for
determining stability via linearization. In general, these methods have three use-
ful features: (I) they are essentially PDE techniques, hence generalize readily to a
wide variety of geometric evolution equations; (II) in contrast to purely geometric
techniques, which generally yield stability theorems only modulo diffeomorphisms,
the methods we consider here prove convergence in normed spaces that allow the
control of diffeomorphisms (which is useful for some applications); and (III) for
quasilinear PDE, these methods can provide a regularity boost- for example, yield-
ing convergence in a c^2 +0: Holder norm for initial data that are close only in a
Cl+f3_norm.
In the second part of the chapter- §3- we survey a few of the dynamic stability
results that have been obtained using linearization and the techniques described in
§1 and §2.
As noted above, in the final part of the chapter- §4- we discuss some (though
not all) dynamic stability results that have been obtained using different notions of
stability, such as variational stability. The idea of variational stability is that if an
evolution equation can be formulated as the gradient flow of a specified functional
F, then the stability of the flow in the neighborhood of a fixed point u of that flow
can be studied by examining the behavior of F near u. Specifically, if u is a lo cal
minimum of F, one expects the flow to be stable at u. If, on the other hand, u is
either a lo cal maximum or a saddle point, then one expects instability. One way of
determining the lo cal behavior of Fis to use an infinite-dimensional version of the
second derivative test from calculus. The analysis then focuses on the study of the
second derivative D^2 F(u) of Fat u.
- Linear stability of Ricci flow
DEFINITION 35.l. Let (35.1) be a parabolic PDE system with a fixed point u.
Let L; ~ C denote the spectrum of the (elliptic) linear operator DuA, and let a
denote the projection of L; onto the real axis. We say that u (or, equivalently, the
PDE at u) is
• strictly linearly stable if a c (-oo, 0),
• linearly stable if a c ( -oo, OJ,
- linearly unstable if an (0 , oo) =f. 0.