1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

458 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

1.1. Rays, Busemann functions and half-spaces.

First we review a few notions, some of which have been used earlier in

this book. In this subsection we do not assume the condition of nonnegative

sectional curvature. The reader familiar with Busemann functions may skip

to subsection 1.3, where we discuss the Sharafutdinov retraction theorem

based on the theory of convex functions in the previous appendix.

1.1.l. Busemann functions.

We begin with the following.

DEFINITION I.1 (Space of rays). Given a noncompact Riemannian man-

ifold (Mn, g), we say that a unit speed geodesic "! : [O, oo) -+ M is a ray

if it is minimizing on each finite interval. Let Ray M denote the set of all

rays in M. Given p E M, let Ray M (p) denote the set of all rays "( in M

emanating from p, i.e., "( (0) = p.

An elementary fact is that for any point p E M, there exists a ray "!

emanating from p. Given a ray"(: [O, oo)-+ M, the Busemann function

b 7 : M-+ IR

associated to "( is defined by

(I.1) b 7 ( x) ~ s-+oo lim ( s - d ("! ( s) , x)).

We have the following bounds for the Busemann function in terms of the

distance function (for a proof, see Lemma B.46 of Volume One for example).

LEMMA I.2. If"(: [O, oo)-+ Mn is a ray emanating from p EM, then

(1) (bounded above by the distance function) for all x E M,

(I.2) lb 7 (x)I ::S d (x,p)

and

(2)

(I.3)

(Lipschitz with Lipschitz constant 1) for all x, y EM,

lb 7 (x) - b 7 (y)I ::S d (x, y).

The Busemann function bp : Mn -+ IR associated to a point p E M

is defined by
bp ~ supb 7 ,
'Y
where the supremum is taken over all rays "! emanating from p. Using
Lemma I.2, we can prove the following (for a proof see Lemma B.48 of
Volume One).

LEMMA I.3 (Busemann function bp - distance and Lipschitz). For any

p E Mn we have for all x E M,

lbp (x)I ::S d (x,p)

and for all x,y EM,

lbp (x) - bp (y)I ::S d (x, y).