1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

46 28. SPECIAL ANCIENT SOLUTIONS

2.4. Classification of asymptotic cones of 3-dimensional K-solutions.

Recall that 3-dimensional singularity models are K-solutions. The following

result gives a classification of the asymptotic cones of noncompact 3-dimensional

K-solutions. In particular, it gives an affirmative answer to Optimistic Conjecture

28. 18 in dimension 3.

THEOREM 28.30 (Asymptotic cones of 3-dimensional K-solutions). The asymp-

totic cone of a noncompact orientable 3-dimensional K-solution must be either a

line or a half-line.

PROOF. STEP 1. Reduction of the problem. Let (M^3 ,g(t)), t E (-oo,0],

be a noncompact orientable 3-dimensional K-solut ion. Since g (t) has nonnegative

curvature operator, by the strong maximum principle for Rm (see Theorem 12 .5 3

in Part II) and by the classification of 2-dimensional K-solutions (see Corollary 9. 19

in [77]), (M^3 ,g(t)) is isometric to either

(1) 52 x IR or (5^2 x !R)/Z 2 , where 52 is the shrinking round 2-sphere and Z 2

is generated by t he isometry (x, y) rl (-x, -y) or

(2) a noncomp act K-solution with positive section al curvature, where M^3 is

diffeomorphic to JR^3.

In case (1), the asymptotic cone of (M^3 , g (t)) is either a line or a half-line

(exercise).
A well-known fact is that for any complete noncompact Riemannian manifold
with nonnegative sectional curvature, the asymptotic cone exists (see Theorem I.26
in Part III for example). The rest of t he proof is devoted to showing that in case
(2) the asymptotic cone is a half-line. We first recall some basic facts.
Let Ray M ( 0) denote the space of (unit speed) rays emanating from 0 in
(M^3 , g (0)) and let d 0 ~ d 9 (o). Recall that a pseudo-metric d 00 on R ay M ( 0) is
defined by (I.7) in Part III; i.e.,

(28.22) d 00 (1'1,12)~ s,tlim -';oo 11i(s)012(t)

for /1, 12 E Ray M ( 0), where the Euclidean comparison angle 1 is defined by

( 28. 23 ) / _,_ _^1 do (x , y)

2

+do (y, z)

2

• do (x, z)
  2
  [O ]
  c,_xy z ...,... cos 2 d ( ) d ( ) E , ?T
  0 x,y 0 y,z
  for x,y,z EM. Recall also t hat the asymptotic cone of (M,g(O)) is isometric

to the Euclidean metric cone Cone (M ( oo), d 00 ) , where (M ( oo), d 00 ) is the

quotient metric space induced by (RayM (0) ,d 00 ) (see Theorem I.26 in Part III).

Thus, the conclusion that the asymptotic cone of (M, g(O)) is a half-line (and hence

the theorem) shall follow from showing that

(28.24) d 00 ( /1, 12) = 0

for all 11,/2 E RayM (0).
Now, by (I.8) in Part III, we h ave that

1. do (1'1 (at) ,/2 (bt)) (^2 + b^2 2 b (d- ( )))

(^1) / 2
t-+oo im t - a a cos^00 11 ,^12.
Hence the desired equality d 00 ( 11 , 12 ) = 0 is true if and only if
(28.25) lim do (1'1 (t) '12 (t)) = 0.
t-+oo t