1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 445

This implies</>' (s) = 1 a.e. Hence, by Lemma H.3 we have.

8

</>' (s) ds = 1

8

ds = s - a,

so that </> ( s) = s as desired.

We now prove the claimed equality (H.36). Since f is concave, for any

p E int (C) such that f (p) <!sup, by Lemma H.31 we have for U in (H.31)^12

(Du f) (p) = (Dmaxf) (p) > 0.

Then, by applying (H.19) to -f, we have

(H.37) D vtCPl f = 1.

1Vf(p)l^2

By the definition (H.8) of directional derivative, for s E [a, b),

f ( exp'Yx(s) (V)) - f bx (s)) = Dv f + o (IVI)

as IVI -+ 0. As a special case of this, we have^13

J bx (s + O")) - f bx (s)) = Dexp:;x\s)('Yx(s+u)/ + O (O") •

Thus

_!!:_</> (s) = lim f bx (s + O")) - f bx (s))

ds+ u-tO+ O"

= O"-tO+ lim D exp'Yx(s) -1 c 'Yx c s+o-ll f

-~~--O"

(H.38) = D exp-l( )('Yx(s+o-)) J,

limo--+O+ 'Yx s o-

where the last equality follows from the fact that Dv f ('"'ix ( s)) is a continuous

function of V. · ·

It follows from Lemma H.41 and (H.34) that for s E (a, b),

( ) 1. exp.=;x

1

(s)hx(s+O"))_(") ()·- \Jf.( .())

H.39 lm - '"'ix + S - 2 '"'/x S.

O"-tO+ (T. j\J f I

Now (H.37) with p ='"'ix (s) and (H.38) imply

d

ds+<f>(s) · = D~(')'x(s)/ IV fl = 1.

0

In view of (H.39), the following should be true.

PROBLEM H.46 (Right continuity of \J f / J\J Jl^2 along its integral curves).

Let f E l!:Wo (C). Show that the vector field \J f (and hence \J f / l\J fl^2 ) is
right continuous along the integral curves to \J f / l\J fl^2. ·

(^12) Indeed, by Lemma H.31, if p E int (C) is such that (Dma:xf) (p) :S 0, then f (p) =
fsup· 13
Note that 'Yx (s) E int (C) for s E (a, b).