1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

76 29. COMPACT^2 -DIMENSIONAL ANCIENT SOLUTIONS

Now we consider some other Cheeger- Gromov limits. Let ti --* -oo and define

vfR(s, t ) = vKR(s +Si, t +ti) and gfR = (vfR)-^1 9cyl· Note that, geometrically,

gJ<R(t) = 'I!i§KR(t +ti), where \J!i : IR x 51 --* IR x 51 is the translation defined by

\J!i(s, B) = (s +Si, e). Then (withμ= 1)

vfR(s, t) = - csch( 4(t +ti )) cosh(2(s + si)) - coth( 4(t +ti)).

Let v~R(s, t) ~ limi-HXl vJ<R(s, t) and g~R ~ (v~R)-^1 gcyl· Assume for simplicity
that si ::; 0 for all i. Let

(29.22) Cleo ~ hm (2ti - Si)·

i-teo

Then

(29.23) ,;~R(,, t) ~ { ~+ e'(HJ•~)-"

To summarize, we h ave the following.

if Cleo= -oo,

if Cleo E IR,

if (J eo = ()().

LEMMA 29.5. Corresponding t o the limiting value of 2ti - Si, we have three

different limits g~R(t) based at ((si, Bi), ti):

(1) If Cleo = -oo, then the Cheeger-Gromov limit is a fiat cylinder.

(2) If Cleo E IR, then the Cheeger-Gromov limit is the cigar so liton.

(3) If Cleo = oo, then the limit vanishes.

4. A priori estimates for the pressure function

Let g(t), t E (-oo, O), be a maximal ancient solution to the Ricci flow on 52.
In this section we derive estimates for the pressure function v( t) defined in (29.4)
and its first few derivatives. Since various geometric quantities may not converge
even in c^0 as t --* -oo, in order to obtain uniform estimates on the time interval
( -oo, -1 J we need to co nsider appropriate quantities. With the aid of these uniform

estimates we shall understand the backward limits of the solution g(t) as t --* -oo.

The choices of quantities which we now consider are affected by th e form of the
nonlinear parabolic equ ation (29.7). In what follows, C denotes a finite constant
which may depend on t h e solution v. When C also depends on other parameters,
such asp E [1, oo) and a E (0, 1), we write C(p) and C(a), respectively.
By the trace Harnack estimate (28.46), ~~ 2 0. Hence the scala r curvature is
uniformly bounded backward in time:

(29.24) sup R < oo.

S^2 x(-eo,-l]

Since ~~ = R v > 0, we obtain on 52 x (-oo, -1] t he uniform bound

(29.25) 0 < v::; C.

We shall derive a number of elliptic inequalities and identities for quantities
involving at most three derivatives of v. The main estimates we shall derive from
these a re (29.28) for !Iv (t)llw2.vcs2) ::; C(p) and (29.38) for jjv\7^2 vllc°'(S 2 ) ::; C(a),
both given b elow. In the following, all of the Sobolev and Holder spaces are defined
using g52.