1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. A POINT PICKING METHOD is5

We call the above properties the 'requirements of (x 0 , to)'. Essentially, (22.5)

says that we have a good relative curvature bound in a parabolic cylinder

centered at (xo, to).

Now we give a proof of the above claim. If the point (xo, t 0 ) itself

satisfies the requirements of (xo, to), then we are done. Otherwise, there

exists (xi, ti) E P(xo, to, (cQ)-i/^2 , -(cQ)-i) such that

Qi~ I Rm J(xi, ti) > 2Q,

so that in particular, (xi, ti) is an a-large curvature point.^2 Note that by

(22.4), we have Qi > ~~ and

ti> - to - HQ-i >~to> - 4 - 8 ~T.

Now if (xi, ti) satisfies the requirements of (xo, to), then we are done.

Otherwise, we can find (x2, t2) E P(xi, ti, (cQi)-^1!^2 , -(cQi)-i) such that

Q2 ~I Rm J(x2, t2) > 2Qi > 2^2 Q.

Note that

5T

t2 2 ti - (cQi)-i 2 ti - HQ]"i 2 to - HQ-i - HQ-i2-i 2

16

.

Moreover, we have the fact that (x2, t2) is an a-large curvature point.^3

If (x 2 , t 2 ) satisfies the requirements of (xo, to), then we are done. Oth-

erwise we repeat the process. Notice that this process must stop at some

point since SUPMx[O,T] I Rm J(x, t) < oo. The last point (xj, tj) satisfies the

requirements of (x 0 , to) with to 2 to - 2HQ-i 2 '.f-'.?[ = '.?[ (more precisely,

tj 2 '.f (1+2-j)).

EXERCISE 22.2. Fill in the details of the above argument to confirm the

existence of the desired point (xo, to).

In conclusion, we have proved the following general point picking result.

LEMMA 22.3 (Large curvature point with local curvature control). Given

a E (0, oo) and c E (0, 1), if there exists a point (xo, to) EM x [T /2, T] with

I Rm J(xo,to) >max { ~, :E}'

then there exists a point ( xo, to) E M x [T / 4, T] with

Q~ A JRmJ(xo,to) A >max {a to'Tc 8 }

(^2) Indeed, JRmJ(x 1 ,t 1 ) > 2JRmJ(xo,to) > ;~ ~ fi:-·
(^3) We have
2cx ex
J Rm J(x2, t2) > 2J Rm J(x1, t1) > t; ~ "t;
(since t2 ~ t1 - HQ-^1 r^1 = t1 - ft ~ ~ti).