1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

6. THE fl;-GAP THEOREM FOR 3-DIMENSIONAL fl;-SOLUTIONS 117

Hence it follows from the strong maximum principle that the round

cylinder 52 x JR^1 and its Z 2 -quotient under the action (p, u) ---+ (-p, -u) are

the only orientable noncompact gradient shrinking solitons which are also

,,;-solutions. (See also Theorem 9.79 in [45] for example.)

The following ,,;-gap theorem for 3-dimensional noncompact ,,;-solutions

was proved by Perelman in §11.9 of [152]. The version we state here is a
slight extension.

THEOREM 19.56 (,,;-gap theorem for 3-dimensional nonspherical space

form ancient 11;-solutions). There exists a universal constant 11;0 > 0 such that

any 3-dimensional nonspherical space form 11;-solution (M^3 , g (t)), t::::; 0, is

actually 11;0-noncollapsed at all scales, i.e., (M, g (t)) is a 11;0-solution.
By a nonspherical space form 11;-solution we mean a 11;-solution which
is not isometric to a shrinking spherical space form.

REMARK 19.57. The nonspherical space form assumption in the above
theorem is necessary. This may be seen from considering the spherical space

forms 53 /r, where r c SO (4), since the order of r is unbounded. Conse-

quently, (noncompact) counterexamples to this theorem in dimension 4 are
given by 53 /r x JR (compare with Example 19.10).

PROBLEM 19.58 (11;-gap theorem for 4-dimensional noncompact ancient
11;-solutions with positive curvature operator). Does the 4-dimensional ver-
sion of the 11;-gap theorem hold, where the nonspherical space form assump-
tion is replaced by the assumption that the ancient 11;-solution is not isomet-
ric to a shrinking 53 /r x JR? It is unclear to us whether or not to expect
a counterexample. Note that complete noncompact Riemannian manifolds
with positive sectional curvature are diffeomorphic to Euclidean space.

We now give the

PROOF OF THEOREM 19.56. Since we can shift time, it suffices to prove

that there exists 11; 0 > 0 such that any 3-dimensional nonspherical space form

11;-solution (M^3 , g ( t)) , t E ( -oo, OJ, is 11;0-noncollapsed at all scales at t = 0.

STEP 1. Any asymptotic shrinking soliton is a round cylinder or its
Z2-quotient.

Let g ( T) ~ g ( -T), T E [O, oo), be the corresponding backward solution
to the Ricci fl.ow and define

fJr(B) ~ T-^1 · g(TB), where BE [O, oo),

for any T E ( 0, oo). Let f : M x ( 0, oo) ---+ JR be the reduced distance function
defined by the backward solution g (T) with basepoint (p, 0), where p EM
is to be chosen later.
By Theorem 19.53, there exist sequences Ti ---+ oo and qi = q (Ti) E M
with reduced distance
3
f, (q· i, T,·) i < -- 2