416 8. APPLICATIONS OF THE REDUCED DISTANCE
4.6. The limit is nonflat. We argue by contradiction. If goo(B) is fiat
for some Bo, then R 900 (8o) = 0 and we get from (8.54) that
1
(8.55) 'Vi'Vj£oo (q, Bo)= 2Bo (goo\j (Bo)·Taking the trace of the above equation, we get /J,. 900 (8o/oo (q, Bo)=~· Plug-
ging this into
we obtain
(8.56) l'V 9=(8o/oo 12 9 00(80) (q, Bo ) = £00 (q,Bo) Bo ·It follows from Lemma 7.59 that (one estimates £i and takes the limit
as i--+ oo)
1 2 nCoBo
£00 (q, Bo) ::::: 4Boe2Co8o d9oa(O) (po, q) - -3-
so that £ 00 (q, Bo) must have a minimum point Pl E M 00. On the other hand,
(8.55) implies that £ 00 (q, Bo) is a strictly convex function on (Moo, goo( Bo)).
It is well known that a strictly convex function has at most one critical
point.^6 Hence Pl is the only critical point of £ 00 (·, Bo) on M 00 , so that
M~ is diffeomorphic to ffi.n. Since, by assumption, (M 00 ,g 00 (Bo)) is a fiat
manifold, it is then isometric to (ffi.n, gE) under ~. We can choose ~ so that
~ (p1) = 0 and global coordinates {xi} such that g 00 (Bo) = (dx^1 )
2
- · · ·
- (dxn)^2. Then (8.55) implies
foo (x, Bo)=
4~
0(x^1 )2
+ · · · +
4~
0(xn)2 + cix^1 + · · · + CnXn + Cn+l·
Equation (8.56) implies (ci)^2 + · · · + (cn)^2 = B 01 cn+l and hence
£00 ( x, 1 ) = 1 (^1 )2 1 ( n^2
4 Bo x + 2Boci + · · · + 4 Bo x + 2Bocn).
From the definition of V 00 (B) = JM= (47rB)-~ e-£=(q,^8 )dμ 900 (8)(q), wecompute V 00 (Bo) = 1 by using the formulas for £ 00 (x, Bo) and g 00 (Bo). This
contradicts V 00 (Bo) = V(oo) < 1 in (8.51). Hence g 00 (B) is not fiat.
PROPOSITION 8.39. In dimension 3 this shrinking gradient soliton limit
has bounded sectional curvature.
(^6) Suppose e= ( q, Bo) has two different critical points p and p1. Let 'Y ( s) , 0 :S: s :S: so,
be a minimal geodesic, with respect to goa(Bo), from p to p 1. We have by (8.55) that
d I s=so rso d2
0 = d/= ( 'Y ( s) , Bo) s=O = l 0 ds 2 e^00 ( "f ( s) , Bo) ds
ro ra 1
=lo \J^2 eoa ("f
1
(s) d (s)) ds =lo 2() l"f' (s)/;oo(Oo) ds > Q,
which is a contradiction.