1547671870-The_Ricci_Flow__Chow

3. PARABOLICITY OF THE RICCI-DETURCK FLOW 83

(To see this, note that if x > 0, then t < x -^1 implies that x < c^1 .)

3.2. DeTurck's notations. This subsection (which is not needed for
the sequel) describes DeTurck's original formulation of the Ricci- DeTurck
flow.
Define the algebraic Einstein operator

by

1

G (v)ij ~ Vij -

2

(tr 9 v) 9ij,

noting that G takes the Ricci tensor to the Einstein tensor:

1

G (Re)= Rc-

2

Rg.

Recall the divergence o and its formal £^2 adjoint o* introduced in (3.19) and

(3.21), respectively, noting that o* of a 1-form X is just a scalar multiple of

the Lie derivative of g with respect to the vector field metrically dual to X.

Using these operators, we can rewrite the linearization of the Ricci tensor

(3.27) as

where

Thus by recalling the Lichnerowicz Laplacian defined in (3.6), we obtain the

following result.

LEMMA 3.17. The variation of the Ricci tensor has the form

D (Reg) (h) = -~ (b.Lh) - [o* (o [G (h)])].

We now reconsider the Ricci-DeTurck flow (3.33) and rewrite the 1-

forms W (t) by computing at any point p E Mn in a coordinate system

chosen so that (r g(t)) ~j (p) = 0. Since we have