CHAPTER 35
Stability of Ricci Flow
Look out kid. It's something you did. God knows when, but you're <loin' it again.- From "Subterranean Homesick Blues" by Bob Dylan
In the study of evolution equations, the concept of stability arises naturally
and is easy to describe heuristically. Consider a nonlinear second-order evolution
equation(35.la)
(35.lb):tu (x, t) = A(x, t, u, Du, D^2 u) ,
u (x, 0) = uo (x),
for a family of maps u (·, t) : Mn ---+Nm between two manifolds. If (35.1) is well-
posed and uo = u is a fixed point, it is natural to ask whether solutions that start
sufficiently near u in a n appropriate topology exist for all time and converge to u.
More generally, if the solution u(t) starting at some u 0 converges to some u 00 , one
can ask whether solutions which start sufficiently close to u 0 also converge to u 00
or perhaps converge to some other point in a specified set (near u 00 ).
If the structure of (35.1) is sufficiently transparent, these questions can some-
times be answered by identifying and obtaining a priori estimates for key (geomet-
ric) quantities. However, another more general approach is often useful, especially
if such estimates are impractical or otherwise unavailable.
In this approach, answering the questions above involves two steps. In the
first step, one computes the linearization DA(u) of the differential operator on
the right-hand side of (35.la) at the fixed point u and analyzes its spectrum. If,
say, the linearization has pure point spectrum, then its eigenvalues with negative
real part together with their associated eigenfunctions indicate perturbations of
the fixed point that are formally stable, while those eigenvalues with positive real
part indicate unstable perturbations. If the linearization DA(u) has eigenvalues
with the real part bounded above by some c < 0, then we say that the equation
corresponding to A is linearly stable at u (see Definition 35.1).
In the second step, one proves dynamic (i.e., asymptotic) stability by arguing
that the linearization DA(u) actually does determine the asymptotic behavior of
solutions that belong to a sufficiently small neighborhood of u in a well-chosen
function space. One effective set-up for doing this is to construct a qualitative
geometric theory for the dynamics by regarding the PDE as an evolution ODE posed
in an infinite-dimensional setting, essentially by using semigroup theory. Other
approaches can be effective as well; indeed, it is sometimes possible to determine
stability without explicit recourse to the linearized operator. But in any case,
this second step can become markedly more difficult if zero is an eigenvalue of
the linearization. If this happens, at least if the null eigendirection is integrable,
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