1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

482 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

is a smooth invertible map. By originally choosing U sufficiently small (in-

dependent of YI), we may assume that

(I.43) d (<I?2 (y), <I?2 (YI)) ~ 2d (y, YI)

for y E 1io. Moreover

f (y) = f (YI)+ Ji (<I?2 (y)).

On the other hand, it follows from (I.42) and (I.43) that

1

I Ji ( <I?2 (y)) - Ji ( <I?2 (YI)) I ~ -. -d ( <I?2 (y) ' <J?2 (Y1))

Slllc

2

~ -. -d (y, YI)

Slllc

for y E 1io. Hence

2

If (y) - f (YI)I ~ -. -d (y, YI)

Slllc

and therefore f is Lipschitz continuous on 1i n V.

(i) By Lemma I.37, there exists so (p) < oo such that x is a i-regular

point of r for all x E M - B (p, so (p)). Hence S (p, s) is a Lipschitz hyper-

surface for all s > so (p).

(ii) By Lemma I.40, the Lipschitz hypersurface S (p, s) is complete as a

length space.

(iii) The last part of the proposition follows from the definition of topo-

logical end and the assumption that so (p) is large. D

5. Approximate Busemann-Feller theorem

In this section we prove an almost distance-decreasing property of the

nearest point projection map in a small tubular neighborhood of a hyper-

surface in a Riemannian manifold.

Let (Mn, g) be a complete orientable Riemannian manifold with sec-

tional curvature bounded above by a constant K,^2 > 0. Let Jin-ICM be a

complete smooth orientable hypersurface and let v be a choice of a smooth

unit normal vector field to Ji. Then 1i is two-sided in M, i.e., there exists

a smooth embedding

<I?: 1-i x [-1, 1] --t M

such that <J? (x, 0) = x for x E 1-l. Let

O+ ~ <J? (1i x [O, 1)),

O_ ~ <J? (Ji x (-1, OJ),

so that <I? (1i x (-1, 1)) = O+ Un and O+ n n = Ji. Without loss of

generality, we may assume that the sets O+ and n_ are on the sides of v

and -v, respectively.

Let M+ be the set of x EM such that either:

(1) x E 1i or