4 10. WEAK MAXIMUM PRINCIPLES FOR SCALARS, TENSORS, AND SYSTEMS
REMARK
10.4. In Section 3 we shall give another proof of Theorem 10.2
using the spatial maximum function. Note also that the lower bound in
Theorem 10.3 can be proved using Theorem 10.2 with u replaced by — u.
In essence the weak maximum principle bounds certain (sub or super)
solutions to heattype equations by solutions to their associated ODE pro
vided F (ix, i) is locally Lipschitz continuous in u. Via the maximum prin
ciple, the following elementary ODE often arises in the study of geometric
flows.
EXERCISE
10.5 (A common
ODE).
Suppose a continuous function x (t)
0 satisfies the ordinary differential inequality
dt
~
°lX
on [0, T) whenever x Ci, where C\ 0 and a 0. Show that
. . . ..... \
_1
A if
'.(t) m^UC1at)~1/aJC2\ = I
(dot)"1'*
*
if*fea—9i^+
Co if t
da
'
SOLUTION
10.5. We compute
A (x«) =
ax~1^
da
d t
K }
dt ~
whenever
x~a C^~a.
Hence
x(t)"
t t
min{Ciat,C
2

a
} .
We have seen applications of Theorem 10.2 at various places earlier in
this book series. Here we recall three applications and then summarize some
others. Recall the evolution of the scalar curvature under the Ricci flow:
(10.5) ^ = AR + 2
Rc2
AR +
R2.
at n
By the scalar maximum principle, —R is bounded from above by a solution
to the associated ODE: ^ = —  s
2
. In particular, if g (t) is a solution of the
Ricci flow on a closed manifold
Mn
with R c at t = 0, then by applying
Theorem 10.2 to (10.5), we have the following scalar curvature lower
bound:
(10.6) *(,,« ) ^
for all x G A4 and t 0. If c 0, then the maximum time interval [0, T) of
existence satisfies the bound T ^ . On the other hand, independently of
the initial condition we have
(10.7) i j (
M
) _ _
for all x £ M and t e (0, T).
The following calculation, carried out in Volume One for the normalized
Ricci flow, shows a formal similarity between the trace Harnack estimate