- ALMOST ,,;-SOLUTIONS
V' ..!L 88 j = 0 at a point, we have at that point,
By' YV'a (-/\-.^0 0) = ( V'a-0) /\-.+-/\^0 0 ( V'a-. 0)
[)yi or 8yJ [)yi 8r 8yJ 8r [)yi 8yJ
-1 0 0= r oyi /\ oyj
andv EJr a (~ or A ~) oyJ = (v. EJr a or ~) A ~ oyJ + ~ 8r A (v EJr a ~) 8yJ
-1 0 0
= r or /\ oyi.Since null (Rm (g 00 ( 0))) is invariant under parallel translation, we have
null (Rm (g 00 (0))) = A^2 N 00 (b, B)129and so the curvature operator of g 00 (0) is identically zero, which contradicts
Rg 00 (x 00 , 0) = 1. Hence Case B cannot happen.
Case C. ASCR(to) = 0 for some to:::::; 0 with n 2".: 3. In this case a gap-
type splitting theorem yields a 2-dimensional ;;;-solution with ASCR = 0, a
contradiction.
By a result of Petrunin and Tuschmann (see Theorem B on p. 777 of
[157]), we have that the universal cover of (Mn, g (to)) is isometric to lEn-^2 x
(L:^2 ,gB (to)) and that (L:,gB (to)) has zero ASCR.^8 It then follows from
Hamilton's strong maximum principle that the universal cover of (Mn, g (t))
is isometric to lEn-^2 x (L:^2 ,gB (t)), where (L:^2 ,gB (t)) is a ;;;-solution (as in
the proof of Theorem 18.17). From Theorem 19.42, (L:^2 , 9B (to)) is a round
sphere and hence JEn-^2 x (L:^2 ,gB (to)) cannot have zero ASCR. We have a
contradiction. Theorem 20.1 is now proved.
SYNOPSIS. In some sense the main result of this section hinges on dimen-
sion reduction and the fact that a product (pn-1, k) x IR, where (pn-^1 , k)
is a closed Riemannian manifold, has AVR = 0.
(A) The case ASCR(to) = oo enables one to do dimension reduction.(B) The case 0 < ASCR (to) < oo leads to the contradiction of an
asymptotic cone being both nonfiat and fiat.
(C) The case ASCR(to) = 0 yields the product of a surface and Euclidean
space as the universal cover; however orientable ;;;-solutions on surfaces are
round 2-spheres, which implies ASCR( to) > 0 and we have a contradiction.
2. Almost /\;-solutions
In this section we prove two propositions. These propositions are the
consequence of Theorem 20.1 and they shall be used in proving Perelman's
compactness theorem (Theorem 20.9).
(^8) Here m;n- (^2) denotes JRn- (^2) with the flat Euclidean metric.