1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^138) 20. COMPACTNESS OF THE SPACE OF IV-SOLUTIONS
and space-time points (xk, 0) with R 9 k (xk, 0) = 1, there exists a
convergent subsequence

(M%,gk (t) ,xk)-+ (M~,goo (t) ,xoo)

and the limit is also a noncompact K-solution for some K > 0.

(ii) The collection Ui,;>O 9Jt~~~h, i.e., the collection of all 3-dimensional

nonspherical space form K-solutions for some K > 0, is precompact

modulo scaling.

PROOF OF COROLLARY 20.11. (i) By Theorem 19.56, there exists Ko

such that

U

9Jtncpt _ 9Jtncpt

K>0 3,K - 3,KQ '

i.e., the (M%, gk (t)) are all Ko-noncollapsed. Now Corollary 20.10 implies

that 9J1 3 ncpt ,~o C 9J13 ' Ko is compact modulo scaling. Thus there exists a con-

verging subsequence

(M%, gk(t), Xk) -+ (M~, goo (t), Xoo)

such that the limit is a Ko-solution. Since (Mk, gk ( t)) E U i,;>O 9Jt~~t, clearly

the limit manifold M 00 is noncompact.

(ii) By Theorem 19.56, there exists Ko such that

U

9Jtnsph _ 9Jtnsph

K>0 3,K - 3,KQ '

Now Corollary 20.10 implies that 9)1~~~: c 9J13,i,; 0 is compact modulo scaling.

Thus there exists a subsequence which converges

(M%,gk(t),xk)---+ (M~,goo (t) ,xoo)

and for which the limit is a Ko-solution. Note, however, that the limit

solution (M 00 , g 00 (t)) could be spherical (see the following remark). D

EXAMPLE 20.12.

(1) For any sequence of solutions whose elements are pointed n-dimen-

sional Bryant solitons, the only possible pointed limits are the n-

dimensional Bryant soliton itself or a round cylinder sn-l x R; see

Example 19.12 and Exercises 19.13 and 19.14.

(2) It seems clear that one should be able to prove that the only types

of limits of Perelman's K-solution on sn are

(a) the shrinking constant sectional curvature sn,

(b) Perelman's K-solution itself,

( c) the Bryant soliton, and

(d) the round cylinder.

Case (a) corresponds to when the sequence of times approaches the singular-

ity time, case (b) corresponds to when the sequence of times stays bounded

away from both -oo and the singularity time, and cases ( c) and ( d) both cor-

respond to when the sequence of times tends to -oo, depending on whether
or not (respectively) the sequence of spatial points stays a bounded distance
from the tips after rescaling the curvature at the tips to be constant.