1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS 177

Note that, a priori, M 00 may be either compact or noncompact. We

shall rule out both of these possibilities simultaneously.^14

Similarly to (21.45), there exists a subsequence such that the functions

in (21.34b )-(21.34d) converge in C^00 on compact sets to some c= functions

Hi --+Hoo, A --+loo, and Vi --+ Voo,

where

(21.60)

Just as in (21.48), we have

I

A A A 1 1

2

D* v = 2t Re, +'V^900 '\J9= + + -·gA H A

00 00 900 J^00 2t = =

00

(21.61)

:S 0.

Furthermore, again by the strong maximum principle, since f; 00 :S 0, there
exists f E ( -oo, OJ such that

(21.62) v 00 = 0 on M 00 x [f, 0) and f;= < 0 on M 00 X (-oo, f)

(this is analogous to (21.49)).

Claim 4. For the fiat solution g 00 (t) on M~ we have that for all

t' E (-oo, 0)

(21.63) f ,, i;2 voo (t') dμ9=(i') < 0.

}B§oo(i')(x 00 ,(-t) )

Proof of Claim 4. If the claim is false, then there exists t' E (-oo, 0)
such that

{B (- (-t'')l/ 2 ) Voo (£') dμ§oo(t') = 0.

} I': §oo(t') Xoo,

Since f;= :S 0, by the strong maximum principle (21.62),
A A ~
V 00 = 0 on M= x [t, 0).

By (21.61) and (21.59), this implies that (M~, g=(t)), t E [t', O], is a

complete shrinking gradient Ricci soliton with zero curvature:

A A A 1 A/

(21.64) '\!^900 '\7^900 f00 +

2

tg 00 = 0 fort E [t, 0)

and

(21.65) Rm9 00 = 0 fort E (-oo, OJ.

(^14) Although the compact case is even easier to rule out, in the sense that it has a
worse isoperimetric constant than the Euclidean isoperimetric constant. ·'