1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

454 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

PROOF. Note that the existence of partitions {Pk} is guaranteed by

Corollary H.56. By Lemma H.57(1) and (2), the lemma follows from verify-

ing that f (I' 00 (so)) < !sup, ( r 00 ) + exists and that

(H.44) ( I'. ) 00 (so) = V' f (I'^00 (so)) 2 for all so E [a, T).

+ IV'f(I'oo(so))I

Let /so,CF : [O, 1]---+ M be a minimal geodesic joining r 00 (so) tor 00 (so+ a}

By Lemma H.41, equation (H.44) follows from

(H.4 5 ) lim 'Yso,CF(O) = V'f(I'oo(so)) ;

CF-+O+ O" IV'f(I'oo(so))l^2

this is the equality we shall prove next.

By Lemma H.57(3),

1. im sup l'Yso,CF(0), = 1. im sup dc(I'oo(so+O"),I'oo(so))

CF-+O+ (}" CF-+O+ (}"

1

::; JV'f(I'oo(so))I.

This implies that when O" is small, 'i'so;(O) is uniformly bounded in Tr 00 (so)M.

Let O"i ---+ o+ be any sequence such that

(H.46) 'Yso,CFi O"i (0) ---+ v; oo E .Lroo(so) rn M an d IV oo I < - IV'f (I'oo^1 (so))I

(such sequences clearly exist). Note that V 00 E Tr 00 (so)C (we leave the proof

of this as an exercise), which follows from the fact that V' f (I' 00 (so)) is an in-

terior point of Troo(so)c. Hence, using r 00 (so+ O"i) = exproo(so) ( O"i. 'i'so:: (O))'
we have
de ( r 00 (so + O"i) 'exproo(so) (O"i v oo))
--'-------------'-----+ o.
O"i
It follows from the Lipschitz property of f that

Since

we have

II (I' 00 (so + O"i)) - f ( expr oo(so) ( O"i Voo)) I

~-------~-----~---+ 0.

O"i

f (I' oo (so+ O"i)) =so+ O"i = f (I' oo (so))+ O"i,

f ( expr oo(so) ( O"i Voo)) - f (I' oo (so))

------~------ 1 ---+ 0.

This implies Dv 00 f (I' 00 (so))= 1, which in turn implies

1

(H.47) D Voo f (I' oo (so))= -

1

v; I 2: IV' f (I' oo (so))I

JVoo I oo