1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

372 7. THE REDUCED DISTANCE

which contradicts (7.143). This completes the proof of the claim and part

(ii).

(iii) By hypothesis, for every q E B (p, r) there exists a C^2 function

<pq : U ---+ JR( defined on a neighborhood U of q such that <pq (q) = f (q),

f (x) ::::; <p 9 (x) for all x E U, IV'<p 91 (q) ::::; C1, and \7\7<p 9 (q) ::::; C2g (q). In

normal coordinates {xi} centered at p, we have

( 0 ~:~~j - rfj ~~k) (q) = (Hess 9 (<p 9 )ij) (q)::::; C2 (9ij) (q)::::; C'C2 (<\j),

where C' < oo is independent of q E B (p, r). Since l\7<p 91 (q) ::::; C1 and

I rrj I ( q) ::::: C"' where C" < 00 is independent of q' there exists a constant C3

independent of q such that the matrix ( 8 ~~~!j -C3r5ij) is negative definite

at q. Choose 'ljJ -(x) = -C3dg(p) (x,p)^2. Then <pq + 'ljJ -is a local upper barrier

function for f + ;/J at q, and since 8 ~i

2

'/:x 1 = ~2C3r5ij, we have

By repeating the proof of (ii), where we replace the geodesic'°'! in (ii) by any

straight line in Bp (r), we see that (! o exp_;^1 +;/Jo exp_;^1 ) (x) is a convex

function of x E Bp (r).

Finally, since - (f o exp_;^1 +'l/J) is a convex function on Bp ( r) , we have

• (f o exp_;^1 +'l/J) is locally Lipschitz on Bp (r), and hence f is locally Lip-

schitz on B (p, r). D

REMARK 7.123. Let f : B (p, r) ---+ JR( be a continuous function, where

p E M and r < inj (p). Suppose there exist C 1 , C 2 < oo such that for each

q E B (p, r) there is a local upper barrier function <pq for f at q satisfying

IV'<p 91 (q) ::::; C1 and \7\7<p 9 (q) ::::; C2g (q).

(i) By Lemma 7.122(iii), we conclude f (x) satisfies (i), (ii), and (iii) in

Lemma 7.117 for any choice of local coordinate chart x.^17 In particular, let

q E B (p, r) and let x be normal coordinates centered at q (in particular,

x (q) = 0). If D^2 f (0) exists, then we can define the Hessian

\7\7 f (q) =';= D^2 f (0)

and the Laplacian

i).j (q) =';= i).9f (q) =';= trlx=O (D^2 J) (x).

It is clear that (\7\7 J) (X, Y) , i).j E Lfoc (B (p, r)) for any continuous vector
fields X and Y on B (p, r).

(^17) Here we have abused notation. When we write f (q), we treat fas a function on
M, and when we write f ( x), we actually mean f o x-^1.