1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 327

Since (xo, To) is a point where T</> (x) P (x, T) attains a negative mini-

mum, we have the estimate

(25.82) T</> (x) P (x, T) 2: -6

in all of M x [O, f].

2.6. The form of the cutoff function.

From (25. 79) we see that the remaining issue is to obtain a cutoff function

</>: M x [O, T] ---+ [O, 1] such that both ~~-fl</> and l\7$1

2

have upper bounds.

Define a 000 function

(25.83) 'ljJ: [O, oo)---+ [O, 1]

so that

(1) 'ljJ (r) = 1 for r E [O, 1] and 'ljJ (r) = 0 for r E [2,oo),

(2)

(25.84) 'lj.J':::;; 0,

\'lj.J'\2 -

T:::;; C, and l'l/J"I:::;; C,

where () E (0, oo) is a universal constant.

Let

f: M x [O,T]---+ lR+

be a 'distance-like' function to be defined as in §4 of Chapter 26 of this
volume. Given any RE [1, oo), we now assume that</> has the form

(25.85)

Note that

and

(25.86)

\1 </> = 'lj.J' (f I R) \1 f

R

\\1 </>12 = ( 'lj.J' (F) )2 l\l f \2

</> R^2 '1j.J (F)

(25.87) :::;; ~2 \\1f1

2