1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 447

for s E (a, b). By Lemma H.15, for each s E (a, b) there exists a constant

speed geodesic (of ( M, g))

as: [O,l]-+C

with as (0) = "fx (s), as (1) = "(y (s), and

L (as)= de ('Yx (s), "(y (s)) = p (s).

By hypothesis, f (x) = a = f (y). Hence we may apply Lemma H.38 to

obtain

(H.40a) ,{_ (. (O) \J f bx ( S)) ) < 7r

as ' l\J f bx (s))l^2 - 2'

(H.40b) L (-a (1) \J f by (s)) ) < ~.

s 'l\Jf('Yy(s))l^2 -^2

We claim that by the first variation formula for arc length, this implies

dp

ds+ (s) ::::; 0.

To see the claim, consider any smooth vector field V along as with

V (a (0)) = \J f bx (s)) and V (a (1)) = \J f by (s)).
s l\Jf bx (s))l^2 s l\Jf by (s))l^2
Note that 'Yx ( s) , "(y ( s) E int ( C) since s E (a, b). This implies as ( u) E int ( C)
for all u E [O, 1]. Thus there exists a smooth 1-parameter family of paths
j3 8 : [O, 1] -+ C, defined for s E [s, s + c:) for some E > 0, with

/3s (u) =as (u) for u E [O, 1],

/3s (0) = "fx (s) for s E [s, s + c:),

/3s (1) = "(y (s) for s E [s, s + c:),

:

8

j3 81

8

=s (u) = V (as (u)) for u E [O, l].

By the first variation formula,

(H.41)

:§ L (/3s) ls=s = -(/3s (0), V (as (0))) + (/3s (1), V (as (1)))

= -(a, (o), l;;(~,(~jil') + (a, (1), l;;(~;(~;;I')

::::; 0,

where we used (H.40). Since L (/3 8 ) 2 p (s) for s E [s, s+c:) and L (/3s) = p (s),

this implies lfo (s) ::::; 0. D