1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

l. LINEAR STABILITY OF RICCI FLOW 281

In this section, we want to study the linear stability of the Ricci flow. However,
as we have seen in the discussion of short-t ime existence for the Ricci flow in
Chapter 3 of Volume One, the linearization D^2 F( u) is not quite elliptic. Therefore,
we determine the stability of the Ricci flow indirectly, by analyzing the spectrum
of the linearization of the equivalent Ricci- DeTurck flow. (The use of the Ricci-
DeTurck flow to study Ricci flow is in many ways similar to the practice of imposing
the Lorentz gauge on the Yang- Mills equations, the wave gauge on the Einstein
equ ations, or the Bianchi gauge in the elliptic context.^1 )
1.1. Spaces and operators.
To begin, let us fix some notation. The reader may reference this section as
needed.
Given a closed, connected smooth manifold Mn, we denote by S^2 (Mn) the
bundle of symmetric (2, 0)-tensors over Mn, and by S! (Mn) the subset of positive-
definite t ensors. For convenience, we write

and

We denote by AP~ AP (T* M) the bundle of p-forms on Mn and by DP~ c= (AP)
the space of differential p-forms.

A smooth Riemannian metric g is an element of 5!. Given g with volume form

dμ, we let /j = /jg : 52 ~ D^1 denote the ("divergence") map^2

/j: h H Jh = -gij '\l ihjk dxk,

whose formal adjoint under the L^2 inner product,

c ,. ) ~ r c ,. i dμ,

}Mn

is the ( "Killing") map J* = J; : D^1 ~ 52 given by

1 1..

6* : w r--+ 2,Lwug = 2 ('Viwj + 'Vjwi) dx' 0 dx^1.

Here L is the Lie derivative, and wij is the vector field metrically isomorphic to w.

We also denote by /j =/jg the map DP~ DP-l formally adjoint to d: DP ~ DP+1;

the meaning should be clear from the context.
It is well known that 52 with the c=-topology is a Frechet space and that 5! c
52 is an open convex cone. There is a natural right action of the group '.D (Mn)
of smooth diffeomorphisms of Mn on 5!' given by ( h, <p) H <p h. For purposes
of studying geometrically distinct metrics on Mn, we identify diffeomorphically
equivalent metrics and regard 5! as a union of orbits Og under diffeomorphism.
Ebin's Slice Theorem [97] shows that, viewed in this way, 5i has many properties of
an infinite-dimensional manifold. The theorem states that for any metric g, there is
a map x: U ~ '.D (Mn) ofa neighborhood U of gin Og such that (x (<pg)) g = <pg
for all <p g E U, and there is a submanifold r of 5i containing g such that the
map U X r ~ 5! given by ( <p g , ')') H (X ( <p g) ) ')' is a diffeomorphism onto a

neighborhood of g in 5!.

(^1) Note, however, that in the Yang- Mills and Einstein cases, imposing the Lorentz or the wave
gauge involves setting restrictions on the fields and their derivatives without changing the field
equations themselves, while the Ricci-DeTurck fl.ow involves an evolution equation that is different
from Ricci fl.ow itself.
(^2) Note that 8 = -div.