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) and gradient estimates for / :
l/l Ce^, |V/|
2
Ce^,
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