1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. REDUCED VOLUME FOR RICCI FLOW 391

We will use the convention

£Jv (r) ~ 0 for r 2: rv.

We can then write the reduced volume as

(8.19)

We compute the evolution of£ along a minimal £-geodesic 'Yv(r) for

0 :'Sr < rv, where VE TpM· For q = 'Yv(r), r E [0,ry), the function

£(·, ·) is smooth in some small neighborhood of (q, r); hence the following

derivatives of£ at such (q, r) exist. Recall from (7.78) that

r^312 ( R + IXl^2 ) (r) = -K + ~L (q, r),

where K = K (r) is the trace Harnack integral defined by (7.75). Thus

(8.20)

Recall equation (7.88):

and from (7.54) recall that

!£ (q, r) = 'Yv (r) = X (r).

Hence the derivative of the reduced distance along a minimal £-geodesic is
given by

(8.21)

by (8.20).

d f)f,

dr [£ hv (r), r)] = or+ \If,· X

1 e 2

=~ 12 K--+R+IXI

2r T

= -~r-3/2K

2

The following lemma can be viewed as an infinitesimal Bishop-Gromov
volume comparison result for the Ricci fl.ow geometry. The striking part
is that no curvature assumption is needed.

LEMMA 8.16 (Pointwise monotonicity along £-geodesics). Suppose that

(Mn, g ( r)) , r E [O, T] , is a complete solution to the backward Ricci flow

with bounded curvature.

(i) For any VE TpM and 0 < r < rv,

(8.22) d~ [(47rr)-nl^2 e-£('Yv(T),^7 )£Jv(r)] :SO,

where equality holds if equality in (7.139) holds.