1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

346 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

REMARK 26.18.

(1) Roughly speaking, estimate (26.42) is sharper for x nearer to y.

This estimate along the diagonal shall be used to improve the global

upper estimate.

(2) Note that, on the other hand, for a closed manifold M with a fixed

metric g the heat kernel tends to the positive constant Vol-^1 (M, g)

as r-+ oo.

PROOF. Fixing any (y, v) E M x [O, T) and applying the mean value

inequality, i.e., Theorem 25.2, at any center (x,r) EM x (v,T] with radius

(spatial scale) ro = 7 to the solution

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

of

au

Dr (x, r) = (Lig( 7 )u) (x, r) - Q (x, r) u (x, r)

.restricted to M x [v, T], we have

u(x,r)::=:; sup u

P ii ( x,r,-2-,--4-.,/T::::v T-V )

(26.44)

where P-g is defined by (25.7) and C 1 , C2, C3 are as in Theorem 25.2.

On the other hand, since u > 0 and P-g ( x, r, 7, -^7 4v) c M x [v, r],

1 .,/T::::v T-V U (z, (}") dμg (z) d<J"::::; 1T r U (z, (}") dμg (z) d<J"

P9(x,T,- 2 -,-- 4 -) v JM

< C'.-n/21

7

• o e Ci(o--v)d O",
  v
  where for the last inequality we used the fact that for O" E [v, r]

JM r u (z, O") dμ-g (z) ::=:; C-n/2 r

0 JM H (z, O"; y, v) dμg(o-) (z)

< (Jn/2eCi(o--v)

• by Lemma 26.14. Hence (26.44) implies

H(x,r;y,v) = u(x,r)

< 4(j1;12c 0 1 eC2T+^03 ".!7' v J.T ____ eCi(o--v)dO"

• Vol-g B-g (x ' ~) 2 T - v