1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

476 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

for all so E (0, c:) such that (r o a)' (so) exists. Therefore

( r o a) ( s) - r ( x) 2: s cos ( ~ - E)

for alls E [O, c:). D

We leave it to the reader to trace through the proof of the 'only if' part

of the lemma to obtain the following slight improvement (the 'if' part is

obvious from Lemma I.34).

EXERCISE I.35 (Characterization of regular points, II). Show that x E

M - {p} is a regular point of r if and only if there exist V E s:;-^1 such that

for any smooth unit speed path/: [O, c:) --t M, where c: > 0, with 'Y (0) = V,

there exists o E (0, c:] such that

r (r (s)) - r (x) 2: Os

for alls E [O, o).

4.3. Regular points on complete noncompact manifolds with

nonnegative curvature.

Next we discuss regular points on complete noncompact manifolds with

nonnegative sectional curvature. Define the maximum angle function of

r

by

(I.27)

angp: M --t [O, 7r]

angp (x) ~ max{L (V, W): V, WE (Y' *r) (x)}.

Note that angp (x) = 0 for x tt. Cut (p). Clearly, if angp (x) < ~'then xis a

regular point of r.

REMARK I.36. There is a similar definition for the set gradient of the
Busemann function associated to a ray or a point and the corresponding
maximum angle function (see [107]).
The following is the main result .of this subsection.
LEMMA I.37 (Maximum angle function tends to zero). Let (Mn,g) be a
complete noncompact manifold with nonnegative sectional curvature. Given
p EM, we have

(I.28) lim angp (x) = 0.

d(x,p)--+oo

In particular, for any c: E (0, ~], there exists so (p,c:) < oo such that xis an

c:-regular point of r for all x E M - B (p, so (p, c:)).

PROOF. We prove the lemma by contradiction. Suppose that there exist

c E (0, ~) and a sequence of points {xi}~ 1 in M such that

ri ~ d (xi,P) --too

and