1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 431

is closest to expP (s (fVi + (1 - t) V2)). From the pointed Cheeger-Gromov

convergence of (M,>..-^2 g,p) to (rpM,g(p),o) as>..--+ o+, we have as

s--+ o+,

Us (t) - (1-t) = 0 (s),

d (rs (us (t)), expP (s (tVi + (1-t) Vi))) = o (s).

We compute, using the convexity off ors (u), that

ls (tVi + (1 - t) Vi)

f (expP (s (tVi + (1-t) Vi))) - f (p)

s

= f (rs (us (t))) - f (p) + f (expp (s (tVi + (1-t) V2))) - f (rs (us (t)))

s s

< (1 - Us (t)) f (rs (0)) +Us (t) f (rs (1)) - f (p)

s

+~~~~~~~~~~~~-Ld (expp (s (tVi + (1-t) Vi)) ,/s (us (t)))

s

< t (J (rs (0)) - f (p)) + (1-t) (f (rs (1)) - f (p))

s

+ ((1 - t) - Us (t)) (f (rs (0)) - f (rs (1))) + 0 (1)

s

= t (J (expP (sVi)) - f (p)) + (1-t) (f (expP (sVi)) - f (p)) + 0 (l).

s

Taking s --+ o+, this implies that

Dw1+(l-t)V2f (p) :s: tDvif (p) + (1-t) Dvd (p).

(iii) Suppose that (Pi, 1/i) E UpECTpC converges to

(poo, Voo) E LJ TpC.

pEC

For any given E > 0, there exists so > 0 small enough such that

f (expp 00 (soVoo)) - f (Poo) _ (Dvoof) (Poo) :S: E.

so

Since f is a continuous function, there exists io (depending on so) such that

for i 2:: io

If (expPi (so\li)) -f (expp 00 (soVoo))I :S: Eso,

If (Pi) - f (Poo)I :S: ESQ.

Hence we obtain

f (expPi (so\li)) - f (Pi) - (Dvoof) (Poo) :S: 3c.

so