2. PROVING THE GLOBAL ESTIMATES 227
Simply put, in taking the time derivative of a quantity such as I 'V' Ql^2 , one
must take into account the evolution of the metric and its Levi-Civita con-
nection as well as the evolution of the tensor itself.
To avoid a notational quagmire, we adopt the following convention in
the proof. If A and B are two tensors on a Riemannian manifold, we denote
by A B any quantity obtained from A ® B by one or more of these oper-
ations: (1) summation over pairs of matching upper and lower indices, (2)
contraction on upper indices with respect to the metric, (3) contraction on
lower indices with respect to the metric inverse, and (4) multiplication by
constants depending only on n and the ranks of A and B. We also denote
by Ak any k-fold product A··· A. For example, this convention lets us
write the conclusion of Lemma 7.4 in the form
:t 1Rml^2 =~ 1 Rml^2 - 2 l'V' Rml^2 + (Rm)*^3.
PROOF OF THEOREM 7.1. The proof is by complete induction on m.
We first consider the case m = 1. The evolution equation for l'V' Rml^2 is of
the form:t l'V'Rml
2
= 2 ( 'V (:t Rm), 'V'Rm)+ 'V Re* Rm*'V'Rm+ Re* ('V'Rm)*^2.
The standard technique of commuting derivatives and using the second
Bianchi identity shows that for any tensor A, the commutator ['V', ~]A is
given by
['V', ~]A= 'V~A-~'VA= Rm*'V' A + 'VRc*A.
By applying formula ( 6.17) and replacing instances of Re with Rm, we get'V (:t Rm) = 'V (~Rm+ (Rm)*
2
) = ~'V'Rm+Rm*'V'Rm.Because
~ IAl^2 = ~(A, A) = 2 (~A, A) + 2 ('V' A, 'VA)
for any tensor A, we conclude that(7.2) :t l'V' Rml^2 = ~ l'V' Rml^2 - 2 l\7^2 Rml
2
+Rm* ('V' Rm)*2
.There are two obstacles to our deriving a satisfactory estimate for l'V' Rml^2
directly from equation (7.2). The first difficulty is the potentially bad term
Rm* ('V' Rm)*^2 on the right-hand side. The second problem is that our
assumptions give us no control on l'V' Rml^2 at t = 0. To circumvent these
obstacles, we define
F ~ t l'V' Rml^2 + f3 1 Rml^2 ,
where f3 is a constant to be chosen below. The strategy for this choice
is that when t is small, it will allow us to control the bad term we get