1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. SETTING UP THE PROOF BY CONTRADICTION AND POINT PICKING 169

REMARK 21.13. Call Ei the 'size' of the parabolic cylinder, which tends

to 0. The lemma says that in Counterstatement A we may adjust the sizes

and choices of { (Xi, ti)} so that we have curvature control, relative to the

size, inside the parabolic cylinder based at (xoi, 0) while the curvature at the

bad sequence of points (xi, ti) is still large relative to the size. A consequence

of this curvature control is the local bound:

I Rm 9 il(x, t) < 21 Rm 9 il (xi, ti)

whenever t E [~,ET] and d 9 i(t) (x, xoi) :S: EOi· We think of Ei as being the

'right' sizes for the contradiction argument.

PROOF. Suppose that i EN is such that (21.18) does not hold. Define

(21.19) μi ( E) ~ sup ( _ max I Rm 9 i I ( x, t) - - -a 2 2) < oo

tE(O,e:2] xEBgi(t) (xoi,C:Oi) t Eoi

for E E (0,Ei]· (Clearly the supremum in (21.19) is finite and attained.)

For such an i, the function μi (c) is both continuous and monotonically

nondecreasing in E and, since (21.18) does not hold, we also have

Furthermore, for this i, since lime:-+O μi (c) = -oo (because 9i (t) has bounded

curvature on Mi x [o, er]), there exists E~ E (0, Ei] such that

μi (cD = o.

That is, the desired inequality (21.18) holds with Ei replaced by E~ and there

exist t~ E (0, (cD^2 J and x~ E B 9 i(t~) (xoi, Eoi) such that

I Rm 9 il (x~, tD = ~ + ~·

ti Eoi

Thus clearly there exists Xi E B 9 i(tn (xoi, Eoi)

I Rm 9i I ( Xi' ti ') > I a + 2 1 ·

ti Eoi

D

Assume from now on that for Counterstatement A the Ei and (xi, ti)

have been adjusted as in Lemma 21.12.

STEP 2. Further improving the bad sequence of points and times by point

picking.

We shall further use point picking to obtain a new bad sequence of points

and times with curvature control in larger and larger parabolic cylinders, so

that we may obtain a complete limit when dilating about these new points

and times.