1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

360 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

PROOF. Let

u (x, r) ~ H (x, r; y, v).

By the Li-Yau inequality (25.58) with c =!,we have

H(y,r;y,v)

H (x T+v.y v)

' 2 ' '

(26. 75)

u(y,r)

• u(x, T!v)

> -Cn(T--r!v) (~)-

2

n x (-Cod~ (x,y))

• e T+v e p 2 r - T+v
  2 2

= e _cll(T~v) 2 ( --2r )- exp -Co~--

2

n ( - d~ (x,y))

r + v r-v

~ e _C11T 2 2 _^2 n exp ( -Co - d~(x,y)) ,

r-v

where Cu < oo is as in Corollary 25.12. Therefore, for A and c as in

Corollary 26.29 (without loss of generality, assume A~ .J2), we have

H(y,r;y,v)

C11T 2

~ e-Cor=v H x, --;y,v dμg (x)

e--2-2-^2 n 1 - d9(x,y) ( r + v )

VolgBg (y,A/9) B9(y,A~) 2.

• 2

  
  

  e-¥2(-2ne-~ { ( ) H (x, r + v; y, v) dμg (x)

  
  

  
  


• VolgBg y,Ay 9) }B 9 y,A~ 2

ce - C11T 2 2-2n e -GoAz 2

> --------

• Volg Bg (y, A/9)
  const
  ~ Volg Bg (y, Jr - v)'

where we used (26. 72) and the volume comparison theorem to change the

radius of the ball (note that A\/ T2v :=:; Ajfx). 0

The following is Proposition 5.1 and Corollary 5.3 in [26].

THEOREM 26.31 (Lower boundfor the heat kernel). There exists a pos-
itive constant c1 depending only on n, T, and the bounds on JRijJ, JRijJ,
JV7kRijJ, and J.6.RJ and there exists a positive constant c2 depending only on
T and the bound on JRij J such that
(26. 76)

H(x TY v) > c1min{ l .l } e-c1(~x~~l

' ' ' - VolgBg(x,Jr-v)'VolgBg(y,Jr-v) ·