- INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 451
Hence, from the definition of y^8 , we haved (y, ys([V'f(yo)[-e1)) ::; s (1 + c1).
By the choice of U 0 we have d (y, ys([V'f(yo)[-e1)) =de (y, ys([V'f(yo)[-e1)); we
have proved that for any s E [O, !s*J and y E U2 nC,
s(l\7f(yo)l-c-1) > l\7f(yo)l-c-1 > l\7f( )l-c-
de (y, ys([V'f(yo)[-e1)) - 1 + c1 - Yo.
Hence the lernrna holds for S& ~ s* (l\7 f (yo)I - c-1) and U = U2. D
We shall construct the integral curves of \7 f / l\7 fl^2 as the limits of
broken geodesics, which we define below. Given a partitionP = {f (x) =a~ so< s1 < .. ·<Sm= T}
of [f (x), T] with T <!sup, we define
IPI ~ l::;i::;m m.ax (si - Si-1)and we definer (x, P) to be a broken geodesic joining xo, x1, ... 'Xm (minimal
between Xi-1 and Xi for i = 1, ... ,m), where
XO= X,
Xi. _ - (x· i-1 )Si-Bi-1 'i.e., Xi is a point in {y EC: f (y) 2: f (xi-1) +Si - Si-1} closest to Xi-1·
Furthermore, we assume that r (x, P) is parametrized as follows:
r (x, P) : [J (x), T]---+ C,
where
r(x,P)(s) ="fi(s)
and 'Yi : [si-li Si] ---+ C is a constant speed minimal geodesic from Xi-1 to Xi·
By (H.42) we have f (xi) = Si·
As we shall see below, when IPI is small, the paths r (x, P) approximate
the integral curves for \7 f / l\7 fl^2.
Note that the proof of the following is the only place where we use the
assumption that C is compact. All other parts of the proof of Proposition
H.53 work under the assumption that !sup> 0.
LEMMA H.55. Let C, x, and f be as in Proposition H.53. There existsa constant C < oo independent of P but depending on diam (C), !sup, and
T <!sup such that for any broken geodesic r (x, P) ands, s' E [f (x), T] we
have
de (r(x,P) (s') ,r(x,P) (s))::; C Is' - sl.