50 18. GEOMETRIC TOOLS AND POINT PICKING METHODS
then there exist a sequence of points {xi}: 1 in M with d (xi, 0) --too and
sequences Ei --t 0 and ri > 0 such that the balls B (xi, ri) are disjoint and
(18.17) sup R:::; (1 + Ei) R (xi),
(18.18)
(18.19)
B(Xi,Ti)
R (xi) r'f --t oo,d (xi, 0) /ri --t oo.
REMARK 18.11 (For the above sequence, rescaled metrics have boundedRon large balls). In the above theorem, let 9i ~ R (xi) g. Then R 9 i (xi)= 1
and
(18.20) sup R 9 i :::; 1 + Ei,
Bgi (xi,i\)where fi ~ R^112 (xi) ri --too as i --too. In particular, the rescaled metric 9i
is such that its scaiar curvature R 9 i at the center Xi is close to its supremum
over the ball B 9 i (xi, ri), where the radius tends to infinity. Note also that
d^1 ;^2 ( d (xi, 0)
9 i (xi, 0) = R Xi) ri ri --t oo
as i --t oo, so that, even after rescaling, the centers Xi of the balls are far
from the basepoint 0.
In §1 of Chapter 20 we shall prove that any complete noncompact /'i,-
solution with Harnack, as given by Definition 19.27, has ASCR = oo when
n~ 3.
2.2. Point picking when sup R = oo.
The following result is a variant, for complete manifolds with supM R =
oo, of Theorem 18.10. This result enables us to perform dimension reduction
(see subsection 2.4 of this section) in certain cases where the curvature
is unbounded. Generally, one assumes unbounded curvature for the sole
purpose of obtaining a contradiction to prove that the curvature is bounded.
The point picking method here is a bit different than in the bounded
curvature case (Theorem 18.10); the idea is to choose a sequence of points
where the curvatures tend to infinity and then to adjust the choices of points
locally.
PROPOSITION 18.12 (Point picking on noncompact manifolds with un-bounded curvature, version I). Suppose that (Mn, g, 0) is a complete non-
compact pointed Riemannian manifold with
supR = oo.
M
Then there exist a sequence of points {xi}: 1 and a sequence ri E (0, 1] such
that
(1) SUPB(xi,ri) R:::; 4R (xi),