80 19. GEOMETRIC PROPERTIES OF K:-SOLUTIONS
In §6 we also concentrate on dimension 3 and prove the 11;-gap theorem
for 3-dimensional noncompact 11:-solutions. This result of Perelman says that
there exists a universal constant 11;0 > 0 such that any 3-dimensional non-
spherical space form 1.,;-solution is actually a 11;0-solution. The proof of this
theorem exploits the fact that for 3-dimensional noncompact 11;-solutions,
the asymptotic shrinking soliton is a round cylinder (or its Z2-quotient). It
also exploits the connection between a lower bound for the reduced volume
and 11;-noncollapsing.
1. Singularity models and /\;-solutions
In this section we briefly recall Perelman's no local collapsing theorem,
its application to the existence of finite time singularity models, and its
motivation to study 11;-solutions. We also discuss intuitive aspects of a special
11;-solution, namely the Bryant soliton.
1.1. 11;-noncollapsing and existence of singularity models.
First we recall the following definition introduced by Perelman.
DEFINITION 19.1 (11;-noncollapsed Riemannian metric). Let /1; > 0 and
p E (0, oo] be two constants. A Riemannian metric g on a manifold Mn
is said to be 11;-noncollapsed below the scale p if, for any geodesic ball
B(x,r) with r < p satisfying jRm(y)j:::::; r-^2 for ally E B(x,r), we have
that the volume ratio is bounded from below:
(l 9 .l) VolB(x, r) > 11;.
rn -
If g is 11;-noncollapsed below the scale oo, then g is said to be 11;-noncollapsed
at all scales.
A complete solution (Mn, g ( t)), t E I, where I c IR is an interval, to
the Ricci flow, is said to be 11;-noncollapsed below the scale p if for every
t EI, g (t) is 11;-noncollapsed below the scale p.
A main relevance of Definition 19.1 to the Ricci fl.ow is Perelman's no
local collapsing theorem (see Theorem 4.1 in [152]).
THEOREM 19.2 (Perelman's no local collapsing). For any finite time
solution of the Ricci flow (Mn,g(t)), t E [O,T), on a closed manifold and
any finite positive scale p, there exists 11; > 0 (where 11; depends only on the
initial metric g ( 0), T, and p) such that the solution is 11;-noncollapsed below
the scale p.
The proof of this result, using the entropy monotonicity formula (17.12),
is discussed in Chapter 6 of Part I. To obtain a local injectivity radius esti-
mate, we recall the following local result in [33] relating volume, curvature,
and injectivity radius (see also Lemma 4.5 of Petersen [155]).
PROPOSITION 19.3 (Cheeger, Gromov, and Taylor). For any constant
11; > 0 and dimension n, there exists a constant io > 0 depending only on 11;
