394 8. APPLICATIONS OF THE REDUCED DISTANCEfor all TE [Ti, T 2 ]). This implies f, is C^00 on M x [Ti, T2] and (8.26) holds on
M x [T 1 , T 2 ]. Thus g ( T) is a shrinking gradient Ricci soliton.
Given To E [T1, T2], by Proposition 1.7, the shrinking gradient Ricci soli-ton structure ( M, g (To) , \7 f, (To) , - ~ ) may be put into a canonical time-
dependent form (1.11) defined for all t <To,g(t) = f (t) <p(t)*g (To),
To
where g(t) is a solution of the Ricci fl.ow, by (1.10), i.e., f (t) = To - t
(s = - ~),and <p(t) is a 1-parameter family of diffeomorphisms with <p (0) =
idM. By the uniqueness of complete solutions of the Ricci flow with bounded
curvature (see Chen and Zhu [82]) and since g(O) = g (To), we have
g ( T) = g (To - T) ,
so that
(8.27) <p(To - T)*g (To)= Tog (T) for TE (0, To].
T
Since !Rm [g (T)]I::::; Co< oo for TE [O, T] (we just use this for T small), by
(8.27) we have
sup !Rm [g (To)]!= sup !Rm [<p(To - T)*g (To)]!
M M
T T
::::; - sup IRm [g (T)]I::::; Co-
To M To
for all T E (0,To]. Hence IRm[g(To)]I = 0. Since O(To) = TpM, Mis
diffeomorphic to IR.n. Part (iii) follows since a fl.at shrinking gradient Ricci
soliton on IR.n must be the Gaussian soliton. D
REMARK 8.18. (i) The Riemannian analogue of Corollary 8.17(i) is
lim VolB (p, r) = 1.
r-+O Wnrn
(ii) Note that for the shrinking gradient Ricci soliton g (T) in subsection
7.3 of Chapter 7, the metric g (0) is not well-defined.
The monotonicity of the reduced volume can be easily generalized to the
following. For any fixed measurable subset Ac TpM, we can define D(A, T)to be the set of vectors V E A such that TV > T, i.e.,
D(A,T) ~{VE A: TV> T} = AnO(T).
It is clear that D(A, T) satisfies D(A, T2) c D(A, T1) if T 2 > T 1.
COROLLARY 8.19 (£-relative volume comparison). Suppose (Mn, g(T)),
T E [O, T], is a complete smooth solution to the backward Ricci flow with the
curvature bound IRm (x, T)I ::::; Co < oo for (x, T) E M x [O, T]. Define for
any T E (0, T) and any measurable subset A c TpM,
VA (T) ~ r (4nT)-nl^2 exp [-f (q, T)] dμg(T) (q).