1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

282 35. STABILITY OF RICCI FLOW

We need only the infinitesimal version of the slice theorem, which gives a useful

decomposition of TgSi. For each g E s i , let 5g ands; be the maps defined above.

If W is a smooth vector field, it is clear that (h, s;w) = (5gh, W), hence that

ker 5g 1-im s;. In fact, these spaces span Tgsi. (See [97] and [25].) One therefore

obtains the orthogonal decomposition

(35.2) TgS:j =Hg EB Vg,

where

Hg~ ker5g and Vg ~ im5;.

Because TgOg = imS; = Vg, this notation is meant to suggest "horizontal" and

"vertical" subspaces.

1.2. Linearization of Ricci-DeTurck flow, revisited.

We work with the linearized Ricci- DeTurck fl.ow with a fixed background metric

g. We use the notation

(35.3) Ag (g) ~ -2 Re (g) - Pg(g),

where

Pg(g)ij ~ -(V'iWj + Y'jWi),

to make explicit the dependence on the background metric g. Here Wis the tensor

field

(35.4) W j -_ ('-1) g jk g g ke pq (" v v9qe A - 2 1,, v e9pq A ).

For smooth initial data g 0 E si, the Ricci-DeTurck fl.ow is then

(35.5a)

(35.5b)

8

8tg =Ag (g)'

g(O) =go.

We observe that if g 0 = g is a fixed point of the Ricci fl.ow, then setting g = g one

has P 9 (g) = 0, so that g is also a fixed point of the Ricci- DeTurck fl.ow. Note that

this is not generally the case for other choices of g.

We have shown in Chapter 3 of Volume One that the unique solution of the

Ricci- DeTurck fl.ow with initial data go is equivalent, modulo diffeomorphisms, to

the unique solution of the Ricci fl.ow with the same initial data. Hence, if we show

that the Ricci-DeTurck fl.ow initial value problem (35.5) is linearly stable for certain

metrics, then linear stability for the Ricci fl.ow initial value problem for these same

metrics readily follows.

The parabolic system (35.5a) is quasilinear, which is important in our imple-

mentation of maximal regularity theory. We can see this explicitly as a consequence

of the following result, which can be easily proved using formulas from Chapter 3

of Volume One.

EXERCISE 35.2. Show that the operator Ag (g) is given in local coordinates by

( ) A ( ) ( , )kepq^8

2

35.6 g g ij =a X, g, g ij

8

XP

8

Xq gke

+ b( x, g, g, A 8 ' g ij )kep 8xPgke^8 + c ( x, g, g, A 8' 82A)ke g ij gk£.

The functions a (x,., ·), b (x , .,., ·), and c (x ,., ·) depend smoothly on x E Mn and

are analytic functions of their remaining arguments.