1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

374 7. THE REDUCED DISTANCE

(iii) A continuous function f : M ---+ R is said to satisfy D.f :::; k in

the viscosity sense for some continuous function k if for every

p E M and any C^2 function <p : U ---+ R on some neighborhood

of p satisfying <p (p) = f (p), f (x) ~ <p (x) for all x E U, we have

D.<p(p):::;k.

(iv) A continuous function f : M ---+ R is called a supersolution of

D.f :::; k for some continuous function k if for every p E M, any

r < i inj (p) , and every 02 function <p on B (p, r) with D.<p = k and

'PlaB(p,r) = flaB(p,r) we have <p:::; f on B (p,r).

Now we can prove the following.

LEMMA 7.125 (Equivalence of notions of supersolution). Let k : M ---+ R

be a continuous function.

(i) Let f: M ---+ R be a continuous function with Hess supp (f) :::; Gp <

oo on B (p,! inj (p)) for each p E M. Suppose that for each q E

B (p,! inj (p)) there is a local upper barrier function <pq for f near

q satisfying l'V'Pql (q) :::; Gp< 00. We have for any <p E c~ (M)

(7.145) JM f D.<pdμ9 :::; JM D.f · <pdμ9.

In particular if D.f :::; k in the support sense, then f satisfies D.f :::;

k in the weak sense.

(ii) f satisfying D.f :::; k in the weak sense is equivalent to f satisfying

D.f :::; k in the viscosity sense; they are both equivalent to f being

a supersolution of D.f :::; k.

PROOF. (i) Let 'ljJ be a 02 function on M. Then having D.f:::; kin the

support sense is equivalent to D. (f + 'ljJ) :::; k + D.'lj.J in the support sense,

and D.f :::; k in the weak sense is equivalent to D. (f + 'ljJ) :::; k + D.'lj.J in the

weak sense. We use a partition of unity {</>a} to rewrite <p = 2-:'P<Pa, so

that we only need to verify inequality (7.145) and (7.144) when <p has small

compact support, say in B (p, !r) for some p E M, where r < min { 1, injJp)}

as determined by Lemma 7.122(iii). From Lemma 7.122(iii) and by adding

to f another concave 02 function if necessary, we may assume that f ( q)

satisfies Hess supp (f) :::; -1 on B (p, r) and that f ( x) is a concave function

on Bp (r) C TpM in normal coordinates {xi} on B (p, r).

From Lemma 7.117(ii) and Lemma 7.119(ii), there are smooth func-

tions fc ( x) such that f c ( x) ---+ f ( x) uniformly on Bp ( 1r) , ( a~:t~j ( x)) ---+

D^2 f (x) for x a.e. on Bp (1r). By Lemma 7.119(iii), (a~:£~j (x)):::; 0 for

all x E Bp (1r). Hence for any 81 > 0 and any 02 -test function <p sup-

. ported in Bp ( !r) , there exist a sequence E:k ---+ O+ and a set W 01 c Bp ( !r)

with meas (W 01 ) :::; 81 such that (

8

f;t ( x)) ---+ D f ( x) and ( ::i~;j ( x)) ---+