1547671870-The_Ricci_Flow__Chow

3. PARABOLICITY OF THE RICCI-DETURCK FLOW 81

parabolic system of partial differential equations. It is a standard result that
for any smooth initial metric 9o , there exists E > 0 depending on 90 such that
a unique smooth solution 9(t) to (3.33) will exist for a short time 0 ::; t < E.
STEP 2. One observes that the one-parameter family of vector fields
W (t) defined by (3.29) exists as long as the solution 9 (t) of (3.33) exists.
Then one defines a 1-parameter family of maps 'Pt : Mn -----+Mn by

(3.35)

[)

at c.pt(p) = - w ('Pt (p) , t)

c.po = idMn.

Notice that 'Pt (p) is constructed by solving a non-autonomous ODE at each
p E Mn. Because Mn is compact, it follows from Lemma 3.15 (below) that
all 'Pt (p) exist and remain diffeomorphisms for as long as the solution 9 (t)
exists, namely for t E [O, E).
STEP 3. One observes that the family of metrics

(3.36) g (t) ~ c.p;9 (t) (0::; t < c)

is a solution to the Ricci fl.ow (3.32). Indeed, we have g (0) = 9 (0) = 90 ,

because c.po = idMn. Then we compute that

it ( c.p; 9 ( t)) = :S I s=O ( 'P;+s9 ( t + S))

= c.p; (%t9(t)) + :sls=O (c.p;+s9(t))

= c.p; (-2Rc (9 (t)) + .Cw(t)9 (t))

• :S ls=O [ (c.ptl o 'Pt+s)* c.p;9 (t)]

= - 2 Re (c.p;9 (t)) + c.p; (.Cw(t)9 (t)) - .C[('P;-1 ). W(t)] (c.p;9 (t))

= -2Rc(c.p;9(t)).

The equality on the penultimate line follows from the identity

~I ('Pt

(^1 0)

'Pt+s) = ('Pt

1

L (~I 'Pt+s) = (c.pt

1

L W (t) ·

us s=O us s=O

Hence g(t) ~ c.p;9(t) is a solution of the Ricci fl.ow fort E [O,c). This

completes the proof of the existence claim in Theorem 3.13.

STEP 4. All that remains is to show that g ( t) ~ c.p; 9 ( t) is the unique

solution of the Ricci fl.ow with initial data g (0) = 90· This will be easier

to do after we have the machinery of the harmonic map heat fl.ow at our

disposal. Consequently, we shall postpone the proof of uniqueness: see Step

4 of the existence and uniqueness proof outlined in Section 4.4.

3.1. Existence of the DeTurck diffeomorphisms. In this subsec-

tion, we establish the existence of a family {'Pt} of diffeomorphisms solving

the ODE (3.35).