1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 449

PROOF. Assuming Lemma H.58 below on the existence of integral curves

(which we shall prove below), we prove that any integral curve can be ex-

tended to a maximal integral curve. Let { si} ~ 0 be a sequence in [a, fsup)

with so= a, Si< si+1, and Si-+ fsup· From Lemma H.58, we can define

1° : [so, s1] -+ C

to be the integral curve of \7 f / l\7 Jl^2 · emanating from 1° (so) = x and for

i 2:: 1 we can inductively define

Ii : [si, Si+i] -+ C
to be the integral curve of \7 f / l\7 Jl^2 emanating from Ii (si) = li-l (si).
We define ix: [a, fsup)-+ C by

ixl[si,si+ 1 J (s) =Ii (s) ·

Clearly, for any b < fsup·, the curve ixl[a,b] is an integral curve of \7 f / l\7 fl^2.

We shall prove that
lim ix (s)
s-+fsup
exists, from which the proposition follows.
Let q be any point in 1-^1 Usup)· By Lemma H.52, we have
ix (s) EB (q, de (x, q)).

Hence there exist a sequence Si -+ !sup and if. E C such that ix (si) -+ if..
It follows from f (ix (si)) =Si that f (if.) =!sup· Applying Lemma H.52 to
de (ix (s), if.), we conclude that ix (s) -+if. ass-+ !sup· This completes the
proof of the proposition. D

Next we turn to the proof of the existence of integral curves, i.e., Lemma
H.58.

4.3.2. The proof of the existence of the integral curve.

In this subsection let C, x, and f be as in Proposition H.53. Define the

convex (superlevel) set

Cu ~ {y E C : f (y) 2:: O"}.

Fix y E Cu and givens E (0, fsup - f (y)] (so that f (y) < f (y) + s :S: fsup),

choose y^8 to be a point in cf(y)+s closest toy (such a point always exists).

Note that

(H.42) f (Y^8 ) = J (y) + S.

The following gives an upper bound for the distance between level sets.

LEMMA H.54 (Distance to a superlevel set). Let Yo EC be a poin:t with

f (yo) < fsup· For c > 0 there exists a neighborhood U of Yo and number

S& > 0 such that for all y E C n U and s E (0, S&) we have

s

d e ( y,y 8 ) 2:: l\7f(yo)l-c;