1547671870-The_Ricci_Flow__Chow

2. EVOLUTION OF THE CURVATURE 109

D
This formula simplifies nicely in the special case of a conformal defor-
mation.

COROLLARY 5.5. If g (t) is a smooth I -parameter family of metrics on
Mn such that
a

atg = f g

for a scalar function f : Mn ___, IR, then

~b,. = - f b. + (~ - 1) \lf. \7.

at 2

In particular, if n = 2, then

a

at b. = -f b. ·

PROOF. It suffices to calculate

gke (gij\li (:tgje) -~\le (gij :tgij)) = (i -~) \lkf.

D
For later use, we also record the following immediate consequence of
Lemma 6.5.

COROLLARY 5.6. If (Mn, g (t)) is a solution of the normalized Ricci flow

a 2r

- g = -2Rc+- g

at n '

where r ( t) denotes the average scalar curvature, then

(5.2)

a

atdμ = (r - R) dμ.

2. Evolution of the curvature

In this section, we compute the PDE for the evolution of the scalar cur-

vature R and study the corresponding ODE obtained formally by ignoring

the Laplacian term. This analysis will let us apply the maximum principle

to obtain a lower bound for R.

LEMMA 5. 7. If g ( t) is a smooth 1-parameter family of metrics on a

Riemannian surface M^2 such that ag /at = f g for a scalar function f , then

a

at R = -b.f - Rf.