1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

448 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

COROLLARY H.50 (Uniqueness of integral curves for V' f /IV' fl^2 ). If x E
C with f ( x) < !sup, then there is at most one integral curve for V' f / IV' f I 2
starting at x.

EXERCISE H.51. Check directly for Example H.34 that Lemma H.49

holds.

Roughly, the following result says that the distance from points on an

integral curve to a fixed point at a 'higher level' is nonincreasing.

LEMMA H.52 (Distance from 'Yx (s) toy EC is nonincreasing). Let C,

x, a, and f be as in Lemma H.45. Let y E C with a < f (y) ~ b. If

"fx : [a, b] --+ C is an integral curve for V' f /IV' fl^2 emanating from x, then
the function
s I-+ de ( "fx ( s) , y)
is nonincreasing for s E [a, b].

PROOF. Let

CT (s) ~de hx (s), y).

By Lemma H.15, for each s E [a, b] there exists a constant speed geodesic

0'. 8 : [O, 1]--+ C with 0'. 8 (0) = "fx (s), 0'. 8 (1) = y, and

L (as)= de hx (s), y) =CT (s).

Since f bx (s)) = s::::; b = f (y), by Lemma H.38,

,{_ (. (O) V' f ("ix ( S)) ) < 7r

O'.s ' IV' f bx (s))l^2 - 2

for s E [a, b]. We leave it as an exercise to show that, by the first variation

formula (which is very similar to the proof of Lemma H.49), this implies

d~~ ( s) ::::; 0 in the sense of the lim sup of forward difference quotients. D

4.3. The existence of maximal integral curves.

In this subsection we prove that the maximal integral curves for the

generalized gradients of concave functions always exist.

4.3.1. The main result.

PROPOSITION H.53 (Existence of maximal integral curve for V' f /IV' fl^2 ).

Let C C (Mn, g) be a compaci1^4 connected locally convex set with nonempty

interior and boundary and let f E <ts.tio (C). If x E C is such that f (x) ~

a< !sup, then there exists a maximal integral curve

"ix : [a, fsup] --+ C

for V' f /IV' fl^2 emanating from x.

(^14) The use of the compactness assumption for C in the proof of Proposition H.53 may
be traced back to Lemma H.55 below.