80 3. SHORT TIME EXISTENCE
Now let f' be a fixed torsion-free connection. (For instance, we could
take f' to be the Levi-Civita connection of a fixed background metric g.)
The considerations above lead us to define a vector field W = W (g, f') by
(3.29) wk = 9 pq (rk pq _ f'k pq ).
Since the difference of two connections is a tensor, W is a globally well-
defined vector field (independent of the coordinates used to describe it lo-
cally). Because W involves one derivative of the metric g, the map
P = P(f') : C^00 (S2T Mn) --t C^00 (S2T Mn)
that corresponds to taking the Lie derivative of g with respect to W, namely
P(g) ~ Cwg,is a second-order partial differential operator. The linearization of P is
(3.30)where Tjk is a linear first-order expression in h. Comparing the equation
above with equation (3.28) leads us to consider a modified Ricci operator,
namely
Q ~ -2Rc+P: C^00 (S2TMn) --t C^00 (S 2 TMn).
From (3.28) and (3.30), the linearization of Q satisfies
DQ(h) = b:. h + U,where Ujk = - 2Sjk + Tjk is a linear first-order expression in h. Hence the
principal symbol of DQ is given by
(3.31) Cf [DQ] (()(h) = 1 (1^2 h,which implies in particular that Q is elliptic.
DeTurck's strategy for constructing a unique short-time solution g (t) of
the Ricci fl.ow
(3.32a)(3.32b)a
Btg = - 2 Re (g),g (0) = 90
on a closed manifold Mn proceeds in four steps:
STEP 1. One defines the Ricci-De'I'urck flow by
(3.33a)(3.33b)a
-gat iJ ·· = -2R· iJ + \i'·W i J + V' J ·W· i ,g (0) =go,where the time-dependent 1-form W is g-dual to the vector field (3.29). In
particular,
(3.34) WJ. -- 9yk. wk _,__ -;-9yk9. pq (rk pq -r-k pq )
depends on g (t), its Levi-Civita connection r (t), and the fixed background
connection f'. It follows from (3.31) that the Ricci-DeTurck fl.ow is a strictly