446 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDSREMARK H.4 7. A nice example to keep in mind is the function given in
Example H.34.The following result says that f is concave along the reparametrized
integral curves for \7 f / I \7f1^2.
LEMMA H.48. Let"/ be an integral curve for \7 f / l\7 Jl^2 and let i' be areparametrization of"/ by arc length. Then the function f o i' is concave.
PROOF. The existence of such a i' follows from Lemma G.5 and the factthat"/ is rectifiable. Now for s1 > so with s1 - so < inj 9 (p), we have (the
first equality follows from Lemma H.45)d:+ (!^0 i') (so)= l\7 fl (i' (so))= Dmaxf
> Dexp:-(I^1 so )(i(s1)/
- d(,:Y(so),,:Y(s1))
> f (i' (s1)) - f (i' (so))
- d(,:Y(so),,:Y(s1))
f(i'(s1))-f(i'(so))
s1 - so
(note that !expi°(~o) (i' (s1)) I = d (i' (so), i' (s1)) > 0 and, since i' is paramet-
rized by arc length, that we have d (i' (so), i' (s1)) ::::; s1 - so). Thus f o i' is
concave (see (H.7) and the sentence thereafter). DThe next result implies the uniqueness of integral curves for \7 f / l\7 fl^2.
Roughly speaking, the result says that the distance between points 'at the
same level' on two integral curves is nonincreasing. The main idea of its
proof is to apply the first variation of arc length formula.
LEMMA H.49 (Monotonicity of distance between integral curves for the
vector field 'v J/l\7 fl^2 ). Let C, x, a, and f be as in Lemma H.45. Let b E
(a, !sup] and let y EC with f (y) =a= f (x). If
"fx : [a, b] -> C and "/y : [a, b] -+ C
are integral curves for \7 f / l\7 !1^2 emanating from x and y, respectively, then
the function
s r-+ de bx (s), "/y (s))is nonincreasing for s E [a, b].
PROOF. Define p : [a, b] -+ [0, oo) by
p ( s) ~ de ("Ix ( s) , "/y ( s)).Since "fxl[a,b) and 'YYl[a,b) are locally Lipschitz, we have that Pl[a,b) is a locally
Lipschitz function. To prove the lemma, it suffices to show that
dp ( ) -'- 1. p ( s + ~s) - p ( s)
d+ s ....,... imsup A ::::; 0
s Dos-to+ us