- NO FINITE TIME LOCAL COLLAPSING 255
Rauch) comparison theorem (comparing (Br-2§ (x, 1), r-^2 9) with the ball
of radius 8 in the unit sphere sn (1)), there exists,.,,=,.,, (n, 8) such that
Vol 9 B (x, r)
n = Volr-29 Br-29 (x, 1) :::'.:: fl,.
r
D
5.1.3. f'i,-noncollapsing in Ricci flow.
DEFINITION 6.55. We say that a complete solution (Mn,g (t)), t E
[O, T), to the Ricci fl.ow, where TE (0, oo], is f'i,-noncollapsed below the
scale p if for every t E [O, T), g (t) is f'i,-noncollapsed below the scale p.
If Mis closed, T1 < oo, and Co~ supMx[O,Ti) IRml < oo, then, using the
metric equivalence e-^2 (n-l)Cog (0) ::; g (t) ::; e^2 (n-l)Cog (0) fort E [O, Ti),^16
we see that for every p E (O,oo) there exists,.,,= ,.,,(n,Co,g(O),T1,p) > 0
such that the solution g (t) is f'i,-noncollapsed below the scale p. Hence we are
interested in f'i,-noncollapsing near T when the solution forms a singularity at
time T. We shall see that when T < oo and M is closed, Perelman's mono-
tonicity of entropy implies that for all p > 0 the solution is f'i,-noncollapsed
below the scale p for some,.,,=,.,, (n, g (0), T, p) > 0.
In §4.1 of [297] Perelman also gave the following.
DEFINITION 6.56 (Locally collapsing solution). Let (Mn,g (t)), t E
[O, T), be a complete solution to the Ricci fl.ow, where T E (0, oo]. The
solution g ( t) is said to be locally collapsing at T if there exists a sequence
of points Xk EM, times tk ---+ T, and radii rk E (0, oo) with rVtk uniformly
bounded (from above) such that the balls·B 9 (tk) (xk, rk) satisfy
(1) (curvature bound comparable to the radius of the ball)
IRm [g (tk)] I ::; r"k^2 in Bg(tk) (xk, rk),
(2) (volume collapse of the ball). Volg(tk) B 9 (tk) (xk, rk)
hm n = 0.
k->oo rk
EXERCISE 6.57. It is interesting to consider solutions to the Ricci fl.ow
which are defined on a time interval of the form (0, T) with the curvature
becoming unbounded as t ---+ O+. For example, consider an initial metric
g 0 on a surface which is C^00 except for a conical singularity. We expect a
smooth solution g (t) of the Ricci fl.ow to exist on some time interval (0, T)
with g (t) ---+ g 0 as t ---+ O+. It is interesting to ask if solutions on closed
manifolds can locally collapse as t ---+ O+. In view of this, for a solution
defined on (0, T), formulate the notion of locally collapsing at time 0.
16For the proof of this metric equivalence, see Corollary 6.50 on p. 204 of Volume
One. The argument there is essentially repeated in the proof of the inequalities in (3.3)
of this volume.