1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

54 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

2.4.l. Point picking criteria for the existence of a line in the limit.
The following result of Perelman in §11.4 and §11.7 of [152] relies on
the Toponogov comparison theorem to construct a line in the limit; see also
Theorem 8.46, especially the argument on p. 319, of [45].

THEOREM 18.17 (Dimension reduction - splitting a line, version I).

Suppose that (Mn,g(t),O), t E (-oo,w), w > 0, is a pointed complete

noncompact ancient solution with bounded Rm 2: 0 such that there exist

sequences Xi EM, ri-+ oo, and Ai-+ oo for which

and

dg(o)(O,xi) >A-

ri - i

(18.24)

Let gi (t) ~ r;^2 g(rrt). If inj(xi,g(O)) 2: lori for some lo > 0 independent

of i, then there exists a subsequence of solutions {(Mn,gi (t) ,xi)} converg-

ing in the C^00 pointed Cheeger-Gromov sense to a complete limit solution

(M~, g 00 (t), x 00 ), t ::::; 0, which is the product of an (n - l)-dimensional

ancient solution with bounded Rm 2: 0 and a line.

REMARK 18.18.

(1) Roughly speaking, the hypotheses of the theorem say that, in a

relative sense, we have a bound on IRml in large balls centered

at points far from the origin which satisfy an injectivity radius

estimate.

(2) Since ri -+ oo, the limit in the theorem is a blow-down limit.

Since in Theorem 18.17 we dlo not know if the limit solution is flat, it is

useful to consider the following variant.

THEOREM 18.19 (Dimension reduction - splitting a line, version II).

Suppose that a complete non compact ancient solution (Mn, g ( t)), t E

(-oo,w), w > 0, with bounded Rm 2: 0 and 0 E M are such that there

exist C < oo and sequences Xi E M and ri E (0, oo) such that

and

d;(o)(O,xi)R(xi,O)-+ oo,

rf R (xi, 0) -+ oo,

(18.25) R(y, 0) ::::; CR (xi, 0) for ally E Bg(O) (xi, ri)·

Let gi (t) R(xi,O)g(R(xi,o)-^1 t). If inj(xi,g(O)) 2: loR(xi,0)-^1!^2 for

some lo > 0 independent of i, then there exists a subsequence of rescaled

solutions {(Mn, gi (t), Xi)} converging in the C^00 pointed Cheeger-Gromov
sense to a complete limit solution (M~, g 00 (t), x 00 ) which is the product of
an ( n - 1 )-dimensional nonfiat solution with bounded Rm 2: 0 and a line.