1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 379

Step 1. We shall show that there exists a constant C 1 < oo depending

only on n and K such that

(26.133) If (x,t)-u(x)I :S C1

for all x EM and t E (0, 1]. By (26.131), this implies

If (x, t) - r (x)I :S C1+1

for all x EM and t E (0, l]. By adding the constant C1 +2 to f (which does

not affect the derivatives off), we obtain (26.128) for any Cn,K 2:: 2C1 + 3.

Using (26.132), JM H (x, y, t) dμ (y) = 1, and l\7ul :S 2, we compute

lf(x,t)-u(x)I :S JMH(x,y,t)lu(y)-u(x)ldμ(y)

(26.134) :S 2 JM H (x, y, t) d (x, y) dμ (y).

(Note that the RHS of (26.134) is independent of the choice of 0.) Clearly,

(26.135) { H (x, y, t) d (x, y) dμ (y) :S { H (x, y, t) dμ (y) = 1.

J B(x,1) J B(x,1)

On the other hand, since Re;::: - (n - 1) K, by Corollary 3.1 of Li and

Yau [121] we have^10

H(x, y, t) :S C1 Vo1-^1 /^2 B ( x, Jt) Vo1-^1 /^2 B (y, Jt) exp ( C 2 Kt - d

2

~' y))

for all x,y EM and t E (O,oo), where C1 < oo is an absolute constant and

where C 2 < oo depends only on n. In particular, if t E (0, 1], then

(26.136)

H (x, y, t) :S C3 Vo1-^1 /^2 B ( x, Jt) Vo1-^1 /^2 B (y, Jt) exp (-d

2

~, y)) ,

where C3 < oo depends only on n and K.
Since B (x, Jt) CB (y, Vt+ d (x, y)), we have fort E (0, 1],

VolB(x,Jt) :SVolB(y,Vt+d(x,y))

(26.137) :S Vol B (y, Jt) C4rn/^2 exp ( C5 ( Jt + d (x, y)))

for some C4 < oo and Cs < oo depending only on n and K, where we
used the following result. Since sect (g) 2:: -K, the Bishop-Gromov volume
comparison theorem implies

VolB(y,Vt+d(x,y)) < VolKB(Vt+d(x,y))

VolB(y, Vt) - VolK B( Vt)

(26.138) :::; C4rn/^2 exp ( C5 (Vt+ d (x, y))),

(^10) Compare with the analogous upper estimate (26.65) for the time-dependent metric
case.