1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

9. WEAK SOLUTION FORMULATION 377

Note that lims---tO+ ftn:n d~ (\l<p · v) dμ = 0 (since n pairs of opposing bound-

ary terms cancel in th~ limit) and

r (vd;. v) <pdμ= 2 r dp (\ldp. v) cpdμ= 2 (1-c) r <pdμ

hr;i hr;i hr;i

since dp \7 dp = (1 - c) v. Hence

lim r (vd;. v) cpdμ= 4 t r <p (x1, ... 'xi-l, 1, xi+l, ... 'xn) dcr.

s---tO+ Jar..n e i=l. }7n-l

We conclude that

2n r <pdμ = r ~d;. <pdμ = r d;. ~<pdμ + 4 r <pdcr.

}7n }7n }7n Jcut(p)

This reflects the fact that, in the sense of weak derivatives, ~d~ ::; 2n and

the distribution [ ~d~] has its singular part supported on the cut locus.

9.5.2. Weak differentiability of the reduced distance function .e. In this

subsection (Mn, g ( r)) , r E [O, T] , is a solution to the backward Ricci fl.ow

with the curvature bound max{JRcJ, JRmJ}::; Co < oo on M x [O, T]. Let

p E M and let .e (x, r) be the .£-distance function defined using basepoint
(p, 0). From the gradient estimate of.£ in space-time and from the Hessian
and Laplacian pointwise estimates of .e in space (Lemma 7.61 and (7.90)),
.£ (·, r) satisfies (7.90) in the support sense from Definition 7.124(i). Lemma
7.125 in the previous subsection applies to .e (x, r) as a function of the space
variable. We have the following.

PROPOSITION 7.128 (Regularity properties of the reduced distance). For

every r E (0, T),

(i) .£ (q, r) is locally Lipschitz in the variable q,

(ii) ~.£ ( q, r) E Lfoc (M) ,

(iii) for any smooth nonnegative function <p on M with compact support,

JM.£(·, r) ~<pdμ::; JM <p~.£ (·, r) dμ,

where dμ denotes the volume form with respect to g ( r).
Note that (i) gives an abstract proof that .e (·, r) is locally Lipschitz,
which is different from the effective proof given earlier.
From Lemma 7.113, we have

JM.£(·, r) ~<pdμ= - JM \7.£ (·, r) · \lcpdμ.

Combining this with the proposition, we obtain that the inequalities (7.91)
and (7.92) hold in the weak sense.
LEMMA 7.129 (Space-time Laplacian comparison for .e holds in weak
sense). Let (Mn, g ( r)) , r E [O, T] , be a solution to the backward Ricci flow
with bounded sectional curvature. We have the following.