1547671870-The_Ricci_Flow__Chow

90 3. SHORT TIME EXISTENCE

is a solution of the Ricci- DeTurck fl.ow (3.33). Indeed, using (3.35) we
compute that

:t9(t) = :t ((<p;-l)*9(t))

= (<t?t

1

r (:t9(t)) + £(\Otl.(it\Ot) [(<p;-l)*9(t)]

= ( <t?t^1 )* ( - 2 Re [9 ( t)]) + £(\Ot). (\O; [W(t)J) [9 ( t) ]

= - 2 Re [9 (t)] + Lw(t) [9 (t)].

But since (3.33) is parabolic, we know that a unique solution 9 (t) does exist.

And once we have 9 (t), we can obtain the diffeomorphisms <ft by solving

the non-autonomous ODE (3.35) at each point, as in Step 2 of DeTurck's

method. So

9 (t) = <p;9 (t)

does exist after all, and solves the Ricci fl.ow.

STEP 4. We now prove that a solution of the Ricci fl.ow is uniquely

determined by its initial data. Suppose 91 ( t) and 92 ( t) are two solutions

of the Ricci fl.ow (3.32) on a common time interval. Let ( <p1)t denote the

solution of the harmonic map fl.ow with respect to 9i (t) and g. Let ( <p2)t

denote the solution of the harmonic map fl.ow with respect to 92 (t) and g.

Then

91 (t) ~ ((<p1)t)*91 (t) and 92 (t) ~ ((<p2)t)*92 (t)

are both solutions of the Ricci-DeTurck fl.ow (3.33). Because 91 (0) = 92 (0)

and (3.33) enjoys unique solutions, we have 91 (t) = 92 (t) for as long as both

exist. But then both ( <p1)t and ( <p2)t are solutions of the ODE

f)

ot (<t?i)t (p) = - W ((<t?i)t (p), t) (i = 1, 2)

generated by the same vector field

wk= 9 pq (r;q -t;q).

Hence (<p1)t = (<p2)t for as long as they are both defined, which implies in

particular that

5. The Ricci flow regarded as a heat equation
   Heuristically, the Ricci fl.ow may be interpreted as a nonlinear heat equa-
   tion for a Riemannian metric. In this section, we develop this point of view,
   which is related to DeTurck's trick. The first step is to consider the Ricci
   tensor in local harmonic coordinates.

DEFINITION 3.28. Local coordinates {xi} are called harmonic coor-

dinates if each coordinate function xi is harmonic:

o -- ~x i -- 9 jk (. a 1 ak - rjkaR. £ ) x i -- -9 jk rjk· i