8 10. WEAK MAXIMUM PRINCIPLES FOR SCALARS, TENSORS, AND SYSTEMS
implies that R 0 at t = 0 (since then (1 3s) R 0), and hence R 0
for t 0. We compute from the evolution equation for a^ = R^ sRgij
that
fcj = 3R (Rij - eRgij) -
Ggpq
(Rip - eRgip) (Rqj - eRgqj)
- WeR (Rij - ERgi:j) + ( 2 ( 1 - s)
|Rc|2
- (l - 3s +
4s2) R2\
g{j.
If at some point aijV^ = 0 for some V with |V"J = 1, then at that point the
eigenvalues of Re are sR, Ai, A2, where Ai + A2 = (1 s) R. Hence
P^V* = 2 (1 - s)
|Rc|2
- (1 - 3s +
4s2) R2
= 2 (1 - s) (Xl +
X22)
- (1 - 3s +
2s2
+
2s3) R2
((1 - s)
3
- (1 - 3s +
2s2
+
2s3)) R2
= e2(l-3e)R2 0
since s 1/3. Thus fiij satisfies the null-eigenvector assumption.
(2) Let otij = \Rgij Rij- Then
±RgiA + Qgpq (lUp - ±R9iA (Rd - \.
Pij = 3i21 Rij -Rg%j ) +
6gpq
I RiP -#7*p ) I Rqj ^^9qj
+ 5R [Rl3 - ^Rgij) -
MRc|2
-
^R2j
9ij.
If at some point otij 0 and
OLIJV^
= 0 for some V with |V| = 1, then at
that point the eigenvalues of Re are |i2, Ai, A2, where Ai + A2 = \R. Hence
ft; W ' = - (jRc|2 - \R2) = 2AiA2.
Since Aa \R = Ai+A2 for a = 1,2, we have Ax, A2 0, so that
(3ijViVj
0
and the null-eigenvector assumption holds.
Finally we mention the matrix Harnack estimate when n = 2, which,
unlike the higher-dimensional case, may be proved using the maximum prin-
ciple for 2-tensors. Let
(M2,g
(*)) be a solution to the Ricci flow on closed
surfaces with positive curvature and define
(10.17) Qi:j = ViVj logR + ^ (R + ^\ gij.
Let Q = g^Qij, which differs from the same named Q in (10.8) by the term
j . From (15.25) ahead we have
d
dt'
(10.18) —Qij = AQtj + 2V log R VQy +
2QikgMQij
- [j+3R)Qij+RQgij.
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