1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

488 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

" ... Gromov has defined, in his lectures [3], the Tits' metric
on the points at infinity, the equivalence classes of rays, of a
Hadamard manifold. Moreover, he has suggested that there
is a counterpart to Tits' metric for nonnegative curvature
and proposed several interesting exercises on such manifolds
(cf. [3], pp. 58-59)."^17
Throughout the discussion in this section, (Mn, g) shall denote a com-
plete noncompact Riemannian manifold with nonnegative sectional curva-
ture. Note that such a manifold either has one topological end or has two
topological ends, and in the latter case it contains a line. To see that M
has at most two topological ends, we argue as follows. If M has at least two
topological ends, then (Mn, g) contains a line, so by the Toponogov splitting

theorem, (Mn,g) is isometric to a product~ x (Nn-l, h), where (Nn-l, h)

is a complete Riemannian manifold with sect 2: 0. If N is compact, then M
has exactly two topological ends. If N is noncompact, then N has exactly

the same number of topological ends as M. By induction on the dimension,

we see that M has exactly two topological ends.

6.1. The mollified distance function.

Fix p EM and s1 E (0, oo). Suppose that the sectional curvatures of

g are bounded from above by ko 2: 1 in the ball B (p, s1 + i). Recall from

(12.31) of Part II that for r (x) ~ d (x,p) the mollified distance functions

re: : B (p, s1) --+ [O, oo)

are defined by

(I.53) re;(x) ~ ]__ r ~ 'T/ ('.1::'.) r (expx (v)) dμg(x) (v)

En JB(o,e:) E

for x E B (p, s1) and E E ( 0, 2 )ko) and where rJ : TxM --+ ~ is the standard

mollifier.^18
By Lemma 12.28 of Part II, re: : B (p, s1)--+ ~is a 000 function. More-
over, by the proof of Lemma 12.29 in Part II, we have the following.

LEMMA I.45 (Convergence of both re: and 'Vre:)· (1) The functions

re: converge uniformly to r as E --+ 0 in any compact subset of M.

(2) In any compact subset of the complement of Cut (p), the gradient

'Vre: converges uniformly to 'Vr as E--+ 0.

By the proof of Lemma 12.30 in Part II, we have for E > 0 sufficiently

small,

(I.54) r - 1 :S re: :Sr+ 1 and l'Vre:I :S C

in B (p, s1), where C is independent of E (see also Lemma 1.5 in [107]).

(^17) (3] in [107] is [9] in the current book, for the reader's convenience.
(^18) Here we isometrically identified Euclidean space En= (JJ.r,gJE) with (TxM,g(x))
(note that 'r/ is rotationally symmetric).