1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

366 7. THE REDUCED DISTANCE

Let U 1 be an open neighborhood of the support of v satisfying U1 CC

U.^16 We can apply Theorem 1 in §6.6.1 on p. 251 of [139] to f and v. This

tells us that for any r:; > 0 there exists a C^1 function f e and a C^1 vector field

Ve = v~ &~i defined on U such that Ve has compact support in U1 and

meas {x E U1 : fe (x)-=/= f (x) or \7 fe (x)-=/= \7 f (x)} :S £,

meas {x E U1 : Ve (x)-=/= v (x) or \7jve (x)-=/= \7jv (x) for some j} :S £,

sup \

0

8

!~ (x)\ :S C(n)Lip(f,U) for all i,

xEV X

sup I

8

8

v~ (x) I :S C (n) Lip (vi, U) for all i, j,

xEV xJ

where meas is the n-dimensional Riemannian (Lebesgue) measure on U and
Lip (f, U) is the Lipschitz constant off on U. By the divergence theorem,
which clearly holds for C^1 functions, we have

r f e div Vedμ9 = - r Ve ' \7 f edμ9 ·

lu1 lu1

Taking r:; '\, 0, we get fu 1 f divvdμ9 = - fu 1 v · \7 fdμ9 and hence

JM f divvdμ9 = - JM v · \7f dμ9.

D

In summary, the divergence theorem for Lipschitz functions follows from
a standard approximation result.

EXERCISE 7.114. Prove Lemma 7.113 in the case where f, instead of v,

has compact support.

9.2. Convex functions. Convex functions have nice differentiability
properties. Recall the notion of derivatives on JR.n.

DEFINITION 7.115 (First and second derivatives). Let U C JR.n be an

open set. Given a continuous function f : U --+ JR., we say

(i) f has first derivative D f (x) E JR.n at x E U if

lf(y)-f(x)-Df(x)·(y-x)l=o(ly-xl) as y--+x.

This definition agrees with Definition 7.108.

(ii) f has second derivative D^2 f (x) E Mnxn at x EU, where Mnxn

is the set of n x n matrices, if there exists a vector D f ( x) E JR.n

(^16) U1 CC U means U1 is compact and U 1 c U. In this case we say that U 1 is compactly
contained in U.