1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^164) 21. PERELMAN'S PSEUDOLOCALITY THEOREM
Now we can define Topping's incomplete solution of the Ricci flow. Let
7f: JR---+ 51 (r) denote the standard covering map given by 7f (x) = [x], the
equivalence class of x mod 27fr.
EXAMPLE 21.6 (Topping's incomplete solution). Let
M2:::::::. (-~ 5' ~) 5 x (-~ 5' 5 ~)
and define the (into) local covering map
¢: M -+ 51 (r) x [-1, 1] c L;^2
by

¢(x,y) = (7r(x), y).

Consider the incomplete solution (M^2 , g'M_ (t)) to the Ricci flow, where

g'M_ (t) ~ rp*gr (t).

Since (L;^2 , gr (t)) converges to a round point as t? Tr, we have

(21.6) lim (inf R 9 r (x, t)) = oo.

t/Tr xEM M

Note also that g'M_ (0) is the standard flat metric on M^2. In particular, for

p E (o, ~),the closed ball B ((0, 0), p) is compact.

THEOREM 21.7 (Topping). There exist counterexamples to Theorem 21.2

if we do not assume that the solution (Mn, g (t)) in its hypothesis is com-

plete.

PROOF. Let (M^2 , g'M_ ( t)) be the incomplete solution in Example 21.6,

let xo = (0, 0), and let ro = !· Note that B (x 0 , ro) is compact. Suppose

that Theorem 21.2 holds for this example and let Eo > 0 be the constant

given in this theorem. By (21.5) we may chooser> 0 small enough so that

Tr <^1 2 ( )2

4

e: 0 = Eoro

(i.e., chooser< li i::6). Then for any TE [!Tr, Tr), by choosing

E = 2fl < 2flr < EQ,

we would have from (21.4)

(21. 7) IRl(xo, t) S -a + 2 4

t Eo

for any t::; (ero)^2 = T. Since T may be chosen arbitrarily close to Tr, by
(21. 7) we have for any t E [!Tr, Tr)

IRI( xo, t ) ::; Tr 2a + Tr^1 = 2a+l Tr.

However this contradicts (21.6). D