1. WEAK MAXIMUM PRINCIPLES FOR SCALARS AND 2-TENSORS 5

on a surface and the scalar curvature lower bound. Let

(M2,g(t))

be a

solution to the unnormalized Ricci flow on closed surfaces with positive

curvature and define

(10.8) Q = AlogR +

R=—\ogR-\V\ogR\2.

We have (compare with (5.38) on p. 145 of Volume One and (10.5) above)

d , , .2

(10.9) — 2 = AQ + 2(Vlogi?,V2 + 2

VVlog R+^Rg

(10.10) AQ +2 (V log R,VQ) +

Q2

and consequently, by Theorem 10.2,

(10.11) Q(x,t)--.

Another application of the scalar maximum principle is the doubling-

time estimate. Recall that

(10.12) —

|Rm|2

A

|Rm|2

+ C (n)

|Rm|3

.

at

Hence if |Rm| K at t = 0, then for all x G M and t 0,

(10.13)

|Rm|2

(x, t)p (t) =

(K'1

- ]-C (n) t\ ,

where p (t) is the solution to the ODE: & = C (n)

p3/2

with p (0) =

K2.

In

particular, if t G 0,(C(n)tf)

- l

, then

(10.14) |Rm|(*)

2

^ -

Some other applications by Hamilton of the scalar maximum principle

in the Ricci flow are

(1) (in dimension 3)

(a) 'Ricci pinching improves' estimate (see the 'second proof of The-

orem 6.30' on pp. 191-194 of Volume One),

(b) 'gradient of the scalar curvature' estimate (see Theorem 6.35 on

p. 194 of Volume One),

(2) (in the Kahler case in all dimensions) defining / by Af = R — r (see

Corollary 2.43, Lemma 2.44, and Corollary 2.45 on p. 79 of Part I of this

volume),

(a) estimating R — r, where r is the spatial average of R, from above

by considering

R-r+\Vf\\

yielding exponential decay when r 0 (i.e., c\

(Mn)

0),

(b) C° and gradient estimates for / :

l/l Ce^, |V/|

2

Ce^,