1547671870-The_Ricci_Flow__Chow

96 4. MAXIMUM PRINCIPLES

THEOREM 4.4 (scalar maximum principle, fourth version: ODE gives
pointwise bounds for PDE). Let u : Mn x [0, T) ---+ IR be a C^2 supersolution
to (4.5) on a closed manifold. Suppose there exists C1 E IR such that that
u (x, 0) 2 C 1 for all x E Mn, and let <pi be the solution to the associated
ordinary differential equation

d<p1 = F ( )

dt <pi

satisfying

Then

u(x,t) 2 <p1(t)

for all x E Mn and t E [O, T) such that <p1 (t) exists.

Similarly, suppose that u is a subsolution to (4.4) and u (x, 0) ~ C2 for all

x E M. Let <p2 (t) be the solution to the initial value problem

d<p2 = F ( )

dt <p^2

<p2 (0) = C2.

Then

u (x, t) ~ <p2(t)

for all x E Mn and t E [O, T) such that <p2 (t) exists.

PROOF. We will just prove the lower bound, since the upper bound is

similar. We compute that

a

at (u - <p1) 2 ~ (u - <p1) + (X, \7 (u - <p1)) + F (u) - F (<p1).

The assumptions on the initial data imply that u - <p1 2 0 at t = 0. We

claim that u - <p 1 2 0 for all t E [O, T). To prove the claim, let T E (0, T)

be given. Since Mn is compact, there exists a constant Cr < oo such that

lu (x, t)I ~CT and l<p1 (t)I ~CT for all (x, t) E Mn x [O, T]. Since Fis locally

Lipschitz, there exists a constant LT < oo such that

IF ( v) - F ( w) I ~ LT Iv - w I

for all v, w E [-CT, CT]· Hence we have

a

at ( u - <p1) 2 ~ ( u - <p1) + ( x' \7 ( u - <p1)) - LT sgn ( u - <p1). ( u - <p1)

on Mn x [0,T], where sgn(-) E {-1,0,1} denotes the signum function.

Applying Proposition 4.3 with /3 ~-LT sgn (u - <p1), we obtain

u - <p1 2 0

on Mn x [O, T]. This proves the claim. The theorem follows, since TE (0, T)

was arbitrary. D

REMARK 4.5. In what follows, we shall freely apply Theorem 4.4 without

explicit reference by simply invoking the (parabolic} maximum principle.