- THE EXISTENCE OF ASYMPTOTIC CONES 465
THEOREM I.25. Let (Mn,g) be a complete noncompact Riemannianmanifold with nonnegative sectional curvature and let p E M. Then there
is a continuous distance-nonincreasing retraction map(I.6) Shar : M -+ Crnax (p).That is, Shar (q) = q for all q E Crnax (p) and
d (Shar (x), Shar (y)) ::; d (x, y)for all x, y E M.
2. The existence of asymptotic cones
The main result of this section is the following. It is used in the proof
of Theorem 20.1 on the ASCR and AVR of A;-solutions.THEOREM I.26 (Asymptotic cone of noncompact manifold with sect 2: 0).
Let (Mn, g) be a complete noncom pact Riemannian manifold with nonnega-
tive sectional curvature. Then the asymptotic cone exists and is a Euclidean
metric cone as in Definition G.29.The proof depends on the notion of ideal boundary; to present this, we
begin with the following. For any ')'1, ')'2 E Ray M (p), we define(I. 7)where 2. is the Euclidean comparison angle defined in (G.17).
This limit exists since, by Lemma G.38, the angle L')'1 (s)pJ'2 (t) is a
nonincreasing function of both s and t.LEMMA I.27 (Pseudo-metric on the space of rays).
(i) For any ')'1, /'2 E Ray M (p) and for any positive numbers a and b,
we have ·(I.8) t~~ d ('Yi(at~, ')'2(bt)) = ( a^2 + b^2 - 2abcos ( d 00 (1'1, ')'2)) )1
12
,where d is the distance function of (M, g).
(ii) d 00 is a pseudo-metric on Ray M (p).PROOF. (i) By the Euclidean law of cosines, we haved^2 ('Yi(at), ')' 2 (bt)) = (at)^2 + (bt)^2 - 2 (at) (bt) cos ( 2. ')'i(at) P/'2 (bt)).
Dividing this by t^2 and taking the limit as t-+ oo, we obtain (I.8).
(ii) It suffices to prove the triangle inequality for d 00 • Let ')'1, ')'2, /'3 E
Ray M (p) and, given a, b, let c be a positive constant to be chosen below.
By dividing the triangle inequality
d ( 'Yl (at) , ')'2 (bt)) ::; d ( 'Yl (at) , /'3 (ct)) + d ( /'3 (ct) , ')'2 ( bt))