1547671870-The_Ricci_Flow__Chow

2. LOCAL VERSUS GLOBAL GEOMETRY 297

(1) 0 ~{VE s;;-^1 Mn: p(V) < oo} is an open subset of s;;-^1 Mn;

(2) Pio : 0 --> (0, oo) is continuous; and

(3) for any V E s;;-^1 M;\o and every L E (O,oo), there exists a

neighborhood U of V in s;;-^1 Mn such that p (U) > L for all U E U.

PROOF. The proof is in two steps.
We shall first show that if a geodesic emanating from p E Mn stops
minimizing at some point, then nearby geodesics emanating from p also
stop minimizing near the same point. If V E 0 and E > 0, the definition of
p (V) implies that

d (p, expP [(p (V) + c) VJ) < p (V) + E:.

Define 8 E (0, oo) by

8 ~ p (V) + E - d (p, expP [(p (V) + c) VJ).

By continuity of the exponential map, there exists a neighborhood Wv of

V in s;;-^1 Mn such that for every W E Wv, one has

8

d(expp[(p(V)+c)W],expp[(p(V)+c)Vl) < 2·

By the triangle inequality, we obtain

0

Id (p, expP [(p (V) + c) Wl) - d (p, expP [(p (V) + c) VJ) I <

2

and thus

0

d (p, expP [(p (V) + c) Wl) < d (p, expP [(p (V) + c) VJ) +

2

8

= p (V) + c -

2

< p (V) + c.

We conclude that p (W) < p (V) + E for all W E Wv. This proves both that

pis upper semicontinuous at V and that Wv ~ 0, hence that 0 is open.

Now suppose that {Vi} is a sequence from sn- l M; such that Vi --> V

but p (Vi) --1-7 p (V). We shall derive a contradiction and thus complete the

proof of the lemma. By passing to a subsequence, we may assume there

is p 00 in the compact set [O, oo] such that p (Vi) --> p 00 -/=- p (V). Since p

is upper semicontinuous at V, we may in fact assume p 00 < p (V), which

implies in particular that expP (p 00 V) is not a cut point of p along the ge-

odesic expP (tV). By passing to a further subsequence, we may by Lemma

B.28 assume either that each expP (p (Vi) Vi) is a singular point of ( expP) *

or else that there are 1/i' E s;;-^1 Mn such that 1/i' -/=- Vi for any i, but

expP (p (Vi) 1/i') = expP (p (Vi) Vi). The first case is impossible, since it im-

plies that ( expPL is singular at p 00 V. In the second case, observe that expP

is an embedding of a sufficiently small neighborhood V of V in TpMn. So

for all large enough i, each 1/i' lies outside V n s;;-^1 Mn. By passing to a