- SPATIAL POINT PICKING METHODS 55
REMARK 18.20. In particular, the hypotheses of the theorem imply that
ASCR (9 (0)) = oo. Note also that the curvature bounds in the balls in
(18.25) are comparable to the curvatures at the centers (this guarantees a
nonfiat limit).PROOF OF THEOREM 18.19. Since Rm 2:: 0 is bounded, by (18.25) and
the trace Harnack estimate, i.e., ~f 2:: 0, we haveBy assumption we also have an injectivity radius estimate so that Hamil-
ton's Cheeger-Gromov-type compactness theorem implies that there ex-
ists a subsequence of {(Mn, 9i (t), xi)} which converges to a limit solution
(M~, 900 (t), xoo)· Since rl R (xi, 0) --+ oo, the metrics 900 (t) are com-
plete with uniformly bounded curvature, R 900 (x 00 , 0) = 1, and Rm 900 2:: 0.
We shall now show that this limit solution is the product of an (n - 1)-
dimensional solution (which must be nonfiat and have bounded Rm 2:: 0)
with a line.
Consider the metric 9 (0) at time zero. By passing to a subsequence, we
may assume that
dg(O)(Xi,Xi+i) 2:: dg(O)(O,xi)and that a sequence of unit speed minimal geodesics /i: [O, dg(O) (0, xi)] --+
M joining 0 to Xi converges to a ray emanating from 0.^8 Let ai denote
a minimal geodesic joining Xi to Xi+!· Since /i converges to a ray, we have
that the angle between /i and /i+l at 0 tends to zero, i.e.,
LxiOXi+i --+ 0.Since the sectional curvatures of 9 (0) are nonnegative, by the triangle ver-
sion of the Toponogov comparison theorem (see Theorem G.33(1) and def-
inition ( G .17) in Appendix G), the Euclidean comparison angle tends to
zero, i.e.,
Lxi0Xi+i --+ 0.Since d 9 (o)(Xi,Xi+i) 2:: dg(o)(O,xi), this implies that the other Euclidean
comparison angle LOxiXi+! tends to 1r.
Now for any p E (0, oo) and for i large enough (depending on p ), there
exist points Pi E /i and qi E ai such that
d 9 (o)(Pi,xi) = d 9 (o)(qi,xi) = pR(xi,0)-^112
since d 9 (o) (xi, Xi+i)R (xi, 0)^1 /^2 2:: d 9 (o) ( 0, Xi)R (xi, 0)^1 /^2 --+ oo. There exists
a subsequence such that -/i and ai converge to rays -100 and aoo (re-
spectively) emanating from x 00 in the limit manifold M 00. The Toponogov
(^8) For the definition of ray, see Definition I.1 in Appendix I.