the reduced volume 3. A weakened no local collapsing theorem via the monotonicity of
- A weakened no local collapsing theorem via the monotonicity
of the reduced volume
In this section, (Mn, g ( t)) , t E [O, T), shall denote a complete solution
to the Ricci fl.ow with T < oo and supxEM, tE[O,ti] IRm9 (x, t)i < oo for any
t1 < T (i.e., the curvatures are bounded, but possibly not uniformly as
t-+ T, as in the case of a singular solution). We fix a time To E (~, T) and
a basepoint Po E M. Let
g ( r) ~ § (To - r).Then (Mn, g ( r)) , T E [O, To], is a solution to the backward Ricci fl.ow with
initial metric g (0) = g (To) and bounded sectional curvature. Let £ (r)
denote the £-length of a curve "'(, let L : M x (0, To] -+ JR denote theL-distance, let .e : M x (0, To] -+ JR denote the reduced distance, and let
V: (O,To]-+ (O,oo) denote the reduced volume, all with respect to g(r)
and the basepoint (po, 0).3.1. A bound of the reduced distance. The following lower bound
of the reduced volume will be used in the proof of the Weakened No Local
Collapsing Theorem 8.26.
LEMMA 8.22 (Lower bound for V at initial time).
(i) (£ upper bound) Fix an arbitrary ro > 0. There exists a constant
C1 > O, depending only on ro, n, T, and SUPMx[O,T/ 2 ] Rc9(t)' and
there exists qo E M such that.e(q,To):::;; C1 for every q E B9(0) (qo,ro).(ii) (V lower bound) Suppose there exist r1 > 0 and v1 > 0 such that
Vol 9 (o) B 9 (o) (w,r1) ~ v1for all w EM. Then there exists a constant C2 > 0, depending only
on ri, v1, n, T, and SUPMx[O,T/ 2 ] Rc9(t)' such thatV (To)~ C2.
PROOF. (i) By Lemma 7.50 there exists qo E M such that^3. n
.e(q 0 , To -T/2) =mm.€ (q, To -T/2):::;; -
2
.
qEMFor any q E Bg(O) ( qo, ro), let {3 : [To - T /2, To] -+ M be a constant speed
minimal geodesic from qo to q with respect tog (0). Defining
Co~ sup Re g(t)'
Mx[O,T/2](^3) This corresponds to time t = T /2.