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)
Rc2
 (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)
Rc2
 (1  3s +
4s2) R2
= 2 (1  s) (Xl +
X22)
 (1  3s +
2s2
+
2s3) R2
((1  s)
3
 (1  3s +
2s2
+
2s3)) R2
= e2(l3e)R2 0
since s 1/3. Thus fiij satisfies the nulleigenvector 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) 
MRc2

^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 ' =  (jRc2  \R2) = 2AiA2.
Since Aa \R = Ai+A2 for a = 1,2, we have Ax, A2 0, so that
(3ijViVj
0
and the nulleigenvector assumption holds.
Finally we mention the matrix Harnack estimate when n = 2, which,
unlike the higherdimensional case, may be proved using the maximum prin
ciple for 2tensors. 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.