1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3SO 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

where VolK B(s) denotes the volume of the ball of radius s in the simply-

connected complete Riemannian manifold SJ( of constant sectional curvature

K.11

Applying (26.137) to (26.136), we have

H (x, y, t) :S C5cn/^4 vo1-^1 B (x, Vi) exp (~C 5 d (x, y)) exp (-d

2

~' y))

(26.139) ::::; C1rn/^4 vo1-^1 B (x, Vi) exp (-d

2

~,y))

fort E (0, 1] and d (x, y) 2:: 1 and where C5 < oo and C7 < oo depend only

on n and K.

Now, using this upper bound for the heat kernel, we compute

(26.140)

2 f H(x,y,t)d(x,y)dμ(y)

}M-B(x,1)

:S2C1rn/^4 yo1-^1 B(x,Vt) { d(x,y)exp(-d

2

(x,y))dμ(y)

j M-B(x,1) 6t

= 2C 7 rn/^4 vo1-^1 B ( x, Vi) 1= sexp (-::) AreaoB (x, s) ds

::::; Csrn/^4 vo1-^1 B(x,Vt)1= exp(-~:) AreaoB(x,s)ds,

where Cs< oo depends only on n and K.

On the other hand, Area8B (x, s) = Js VolB (x, s), so that by integrat-
ing by parts in (26.140) (and using the Bishop-Gromov volume comparison
theorem to control the growth of AreaoB (x, s) and to see that the boundary
term at s = oo vanishes), we obtain for all x EM and t E (0, 1],

2 { H(x,y,t)d(x,y)dμ(y)

}M-B(x,1)

< Cst-n/4 /,=!__exp (-s

2

) VolB (x, s) ds

• 1 4t 8t VolB (x, Vt)

since the boundary term at s = 1 is equal to

• C st -n/4 exp ( --1) VolB (x, 1) < 0.
  8t VolB (x, Vt) -

(^11) Since t :S: 1, we have Volx B( Vt) 2:: c 1 tnl (^2) for some c 1 > 0 depending only on n and
K. Note also that for any r E (O,oo) we have
Volx B(r) :S: Cse^05 ,,.,
where Cs < oo depends only on n and K.