444 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS
4.2. Integral curves for concave functions and their properties.
Recall that ctm-o (C) is defined in (H.30). We denote !sup~ supyEC f (y)
for f E ctm-o (C).
DEFINITION H.43 (Integral curves for generalized gradient \7f/l\7f1^2 ).Suppose x EC and f (x) ~a.
(1) A continuous path(H.34)
"fx : [a, b] -+ Cis an integral curve for (the generalized gradient) \7 f / I \7f1^2
emanating from x if it satisfies the following properties:
(i) "fx (a) = x,
(ii) "fxl[a,b) is locally Lipschitz (and hence rectifiable),
(iii) for alls E [a, b) we have that f bx (s)) < !sup, the right tan-
gent vector bx)+ (s) exists, and. \7 f ("Ix (s))
bx)+ (s) = l\7 f bx (s))l^2 '
where \7f is the unique and nonzero (see Lemma H.36(iv))
generalized gradient off given by (H.17).
(2) We say that such an integral curve is maximal if f (b) =!sup·
Note that if f is smooth, then bx)+(!)= 1.
EXERCISE H.44 (Example H.34 revisited). For C ~ [-1, 1] x [-1, 1] and
the function f (x, y) = d ( (x, y ), 8C) (which is the negative of the function in
(H.24)), what is the maximal integral curve for \7 f / l\7 fl^2 emanating from
(x,y) EC?We prove some properties of integral curves assuming their existence,
which is established in the next subsection.
LEMMA H.45 (! along integral curve for \7 f / l\7 !1^2 ). Let f E ctm-o (C).
Let x E C be a point such that f (x) ~ a < !sup· If b E (a, !sup] and
"Ix : [a, b] -+ C is an integral curve for \7 f / l\7 Jl^2 emanating from x, then
(H.35) f bx (s)) = s
for alls E [a, b].
PROOF. Define
¢ (s) ~ f bx (s)) ·Since ¢ (s) is locally Lipschitz for s E [a, b), its derivative ¢' (s)· exists for
a.e. s E (a, b). We claim that
d·
(H.36) ds+ ¢ (s) = 1 for a.e. s E (a, b).