1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

6. EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY 491

We have

limLe = oo,

e-+0

whereas the length along IR x {O} from (0, 0) to (1, 0) is equal to 1.

Note that we may take the product of this example with Euclidean

space to get higher-dimensional examples. In the above example,

although the limit is smooth (fl.at), the 'Lipschitz constants' of the

curves Ce tend to oo as E --+ 0.

(2) (Less bumpy hypersurfaces limiting to a hyperplane) Let Lx J denote

the greatest integer less than or equal to x. Consider the function

f : IR--+ [-!, !J defined by

f(x) ~Ix-LxJ-~I·

Then f is Lipschitz with Lipschitz constant 1 (f is not smooth

exactly at the half-integers). Consider the Lipschitz plane curves

Se ~ { (x, Ef ( E-^1 x)) : x E IR}

defined for E > 0. We have

lim Se= IR x {O}.

e-+0

On the other hand, the length along Se from (0,0) to (1,Ef (c^1 ))

is equal to J2 independent of E.

6.2. An approach toward the space of points at infinity.

In this subsection we discuss an approach toward constructing the space
of points at infinity.

6.2.1. A presumed result on distance-nonincreasing maps.

This discussion hinges on the validity of the following presumed result

(see Proposition 2.2 in Kasue [107]).

PRESUMED THEOREM I.50 (Distance nonincreasing maps between large
spheres). Suppose (Mn, g) is a complete Riemannian manifold with nonneg-

ative sectional curvature. For any p E M there exists so (p) < oo such that

if t ::::: s ::::: so (p), then there exists a map

</Js,t: S (p, s)--+ S (p, t)

such that ·

(1) ((relatively) distance nonincreasing)

1~ 1~

(I.60) tds(p,t) (</Js,t (x), <Ps,t (y)) :::; -;ds(p,s) (x, y),

where ds(p,s) is defined by (I.15) on the sphere S (p, s); that is, the

map of metric spaces

</Js,t : ( S (p, s) , ~ds(p,s)) --+ ( S (p, t) , ~ds(p,t))