1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. REDUCED VOLUME AT THE SINGULAR T IME FOR TYP E I SOLUTIONS 145

2. Reduced volume at the singular time for Type I solutions

It is well known that for complete solutions of the Ricci flow wit h bounded

curvatures, the reduced volumes h ave the monotonicity property (see Corollary
8 .17 in P art I for an exposition). Using estimates from the previous sect ion, as
well as estimates proved in this section, we shall establish the well-definedness
and monotonicity of t he r educed volume based at t he singula r time for Type I
solutions. In the next section we sha ll use this monotonicit y to obtain shrinking
GRS as singula rity m o dels.
Under the hypotheses of Theorem 30.10 and corresponding to the solut ion

(M~,g 00 (t)) and f. 00 , we d efine the r educe d volume based at the singular

time (Poo, Woo) by

(30.44) Voo (t) ~ VPoo,Woo (t) ~ 1 (47r (woo - t))-n/

2

e- foo(x,t)dμgoo(t) (x).
Moo
We have the following reduced volume monotonicity based at the singular time
for a sequen ce of solutions.

THEOREM 30.13. Assume the hypotheses of Theorem 30.10 with r = 1, i.e.,

a Type I assumption. Let V 00 (t) be a reduced volume based at the singular time

corresponding to f. 00. W e have th e following:

(1) (Bound ed above by 1) V 00 (t) is well defined and V 00 (t) :::; l.

(2) (Monotonicity) V 00 (t) is differentiable and 1ft,V 00 (t) ?:: 0. In particular,

limt--+w 00 V 00 (t) :=:; 1.

(3) (The equa lity case) IfV 00 (t1) = V 00 (t2) for some ti < t2, both in (a 00 , woo),

then (M~, g 00 (t), f. 00 (t), -w~-t) is a shrinking GRS structure for all

t E [ti , t2]; i.e.,

R e () + 'J 9 oo(t)'J 9 oo(t)g (t) - 1 g (t) = 0.

900 t oo 2 (woo - t) oo

Since Theor em 30.10 is stat ed for a sequence of solutions to t he Ricci flow, we
reformulate Theorem 30. 13 for a single Type I solution.

COROLLARY 30.1 4 (Reduced volume monotonicity based at the singular time
for a Type I solut ion). Let (Mn,g(t)), t E [O, T), be a complete Type I singular

solution with T < 00. Corresponding to ep,T in (30.31), let ~J,T (t) be the reduced

volume based at the singular time. Then:

(1) Vp,T (t) is well defined and Vp,T (t):::; l.

(2) Vp,T (t) is differentiable and

d -

dt Vp,T (t) ?:: 0.

In particular, limt--tT Vp,T ( t) :::; 1.

(3) If Vp,T (t 1 ) = Vp,T (t2), where 0 <ti < t2 < T , then

(Mn,g(t) , ep,T (t) ,-T~t)

is a shrinking GRS structure for all t E [t1, t2]; i.e.,

R e g(t) + \l^2 fp,T (t) -^1 (

2 (T _ t) g t) = 0.