- MONOTONICITY OF ENERGY FOR THE RICCI FLOW 199
2.2. Correspondence between solutions of the gradient fl.ow
and solutions of the Ricci fl.ow.
2.2.1. Converting a solution of the gradient flow to a solution of Ricci
flow. We first show that solutions of the gradient fl.ow, if they exist, give
rise to solutions of the Ricci fl.ow with the same initial data (Lemma 5.15).In particular, suppose we have a solution (9 (t), J (t)) of the fl.ow (5.25) and
(5.26) on [O, T]; then we can obtain a solution g( t) of the Ricci fl.ow on [O, T]by modifying 9 (t) by diffeomorphisms generated by the gradient of J (t).
LEMMA 5.15 (Perelman's coupling for Ricci fl.ow). Let (9(t), f(t)) be a
solution of (5.25) and (5.26) on [O, T]. We define a 1-parameter family of
diffeomorphisms w(t): M ~ M by
d -(5.32) dt w(t) = \lg(t)f(t),
(5.33) w(O) = idM.
Then the pullback metric g(t) = w(t)*9(t) and the dilaton f (t) = f o w(t)
satisfy the following system:(5.34) ag = -2 Re
at '
af 2
(5.35) at = -L1f + l\7 fl - R.REMARK 5.16. Basically we can see this from the facts that L'Vf9 =
2\7\7 f and L\1 ff = I \7 f I 2. For the sake of completeness we give the detailed
calculations below.
PROOF. First note that by Lemma 3.15 of Volume One the system of
ODE (5.32)-(5.33) is always solvable. We compute
~; = w ( ~~) + w (L'Vg/9) = -2w* (Re (9)) = -2Rc (g).
To obtain the equation for ~{,we compute
a f = a (! o w) = a J 0 w + / (v !) 0 w aw)
at at at \ 'atg= (-lif-R) ow+ [(vl) ow[;
= -L1f - R + l\7fl^2 ,
where barring a quantity indicates that it corresponds to 9 (t). D
So a solution to the gradient fl.ow (5.25)-(5.26) yields a solution to the
Ricci fl.ow-backward heat equation system (5.34)-(5.35). Note that we can
first solve the Ricci fl.ow (5.34) forward in time and then solve (5.35) back-
ward in time to get a solution of (5.34)-(5.35); this will be useful in appli-
cations.