1547671870-The_Ricci_Flow__Chow

94 4. MAXIMUM PRINCIPLES

THEOREM 4.2 (scalar maximum principle, second version: lower bounds

are preserved for supersolutions). Let g (t) : t E [O, T) be a 1-parameter

family of Riemannian metrics and X (t) : t E [O, T) a 1-parameter family

of vector fields on a closed manifold Mn. Let u : Mn x [O, T) -+ JR be

a C^2 function. Suppose that there exists o: E JR such that u (x, 0) 2': o:

for all x E Mn and that u is a supersolution of the heat equation at any

(x, t) E Mn x [O, T) such that u (x, t) < o:. Then u (x, t) 2': o: for all x E Mn

and t E [O, T).

The idea of the proof is eminently simple: in essence, one just applies

the first and second derivative tests in calculus.

PROOF. If H : Mn x [O, T) -+JR is a C^2 function and (xo, to) is a point

and time where H attains its minimum among all points and earlier times,

namely

then

(4.2a)

(4.2b)

(4.2c)

H (xo, to) = min H,

Mnx[O,to]

aH

at (xo, to) :::;: 0,

V' H (xo, to) = 0,

l::!.H (xo, to) 2': 0.

Consider the function H defined by

H (x, t) ~ [u (x, t) - o:] +ct+ c:,

where c: is any positive number. Note that H 2': c: > 0 at t = 0. Using (4.1),

we find that H satisfies

(4.3)

aH

at 2: l::!.H + (X, V' H) + c:

at any point where u < o:. To prove the theorem, it will suffice to prove the

claim that H > 0 for all t E [O, T). To prove that claim, suppose that H:::;: 0

at some (x1, t1) E Mn x [O, T). Then since Mn is compact and H > 0 at

t = 0, there is a first time to E (0, t1] such that there exists a point x 0 E Mn

such that H (xo, to) = 0. Then since

u (xo, to) = o: - eta - c: < o:,

combining (4.2) with (4.3) implies that

aH

0 2': at (xo, to) 2: f:::.H (xo, to) + (X, V' H) (xo, to) + c: 2: c: > 0.

This contradiction proves the claim and hence the theorem. D