1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

390 8. APPLICATIONS OF THE REDUCED DISTANCE

formulas to be considered rigorously later. Let X =\!£.Again, provided we

can differentiate under the integral sign, we compute

dV Q { D (( 4 )-n/2 -£(·,T)d )

dr - JM dr 1fT e μg(T)

=JM (-2: - ~: + R-l\7£12 + ~e) (41fr)-n/2 e-edμ

::; o,

where we used (7.91) to obtain the last inequality. Note that we actually

have the pointwise inequality

~ ((41fr)-n/^2 e-edμ)

= (- 2 : - ~: + R - l'V£J^2 + ~£) (41fr)-n/^2 e-edμ::; 0.

Pulling this back to the tangent space TpM, we have

d~ [(41fr)-n/2e-e(!'v(T),T)_cJv(r)]::; 0

as in (8.22) below.

2.3. Monotonicity of reduced volume: A proof using the .C-

Jacobian. In this subsection we give a rigorous proof of the reduced volume

monotonicity. Recall that the open set

O(r) = O(p,o)(r) ~{VE TpM: rv > r} C TpM

is given by Definition 7.96(iii) and satisfies n(p,O) ( T2) c n(p,O) ( T1) if T1 < T2.

Recall from (7.136) that the .C-exponential map restricted to O(r),

.CT exp: n(p,O) (r) __, M\.CCut(p,o)(r)

is a diffeomorphism. If n(p,O) ( T) = TpM for some T > 0, then .c Cut(p,O) ( T) =

0 and Mn is diffeomorphic to Euclidean space.

Since .CCut(p,o)(r) has measure zero in (M,g(r)), we have by the defi-

nition of the .C-Jacobian and (7.136),

V (r) = r (41fT)-nf^2 exp[-£ (q, r)] dμg(T) (q)

} M\£ Cut(p,O) ( T)

(8.18) = r (41fT)-nf^2 e-£('Yv(T),T)_cJv(r)dx(V),

ln(p,O)(T)

where 'Yv is the .C-geodesic emanating from p with limT->O+ ..Ji-rv ( T) = V
and CJ v ( T) is the .C-J acobian associated to the .C-geodesic 'YV. Here dx
is the volume form on TpM with respect to the Euclidean metric g(O,p),
.CT exp(V) = 'YV ( T) , and

.CJv (r) dx (V) =(.CT exp)* dμg(Lrexp(V),T)·