1547671870-The_Ricci_Flow__Chow

258 9. TYPE I SINGULARITIES

Namely, large negative sectional curvatures can occur only in the presence
of much larger positive sectional curvatures:

THEOREM 9.4. Let (M^3 ,g(t)) be any solution of the Riccifiow on a

closed 3-manifold for 0 :::; t < T. Let v ( x, t) denote the smallest eigenvalue

of the curvature operator. JfinfxEM3 v (x, 0) 2: -1, then at any point (x, t) E

M^3 x [O, T) where v (x, t) < 0, the scalar curvature is estimated by

R 2: lvl (log lvl +log (1 + t) - 3).

Note that one can always achieve infxEM3 v (x, 0) 2: - 1 simply by scaling

g (0) by a sufficiently large constant. Note also that if v :::; -e^6 , we have

1

R 2: 2 lvl log lvl.

In particular, R » lvl when v « -1. Thus whenever we encounter a large
negative sectional curvature at some point and time (x, t), we find a much
larger positive sectional curvature at the same point and time. As we shall
see, this estimate implies that limits (if they exist) of sequences of dilations
about a finite-time singularity have nonnegative sectional curvature.
Before proving the theorem, we establish two technical lemmas.

LEMMA 9.5. At each fixed time t, the subsets

JC c (!' 2T M3 ®s /\ 2T M3) x

defined by

tr lP' 2: - 3 (1 + t),

JC = lP' and if v (IP') :S: -1 / (1 + t), then

tr IP' 2: Iv (IP')I (log Iv (IP')I +log (1 + t) - 3)

are invariant under parallel translation and are convex in each fiber.

PROOF. Invariance under parallel translation is clear. To prove convex-

ity at each fixed x E M^3 and t E [O, T), first consider the region K c JR^2

given by

v2:-3(1+t),

v 2: - 3u,

K = (u,v)

and if u 2: 1 / ( 1 + t) , then

v 2: u (log u + log ( 1 + t) - 3)

See figure (1) and notice that all three curves shown intersect when u =

1/ (1 + t). The region K is bounded on the left by the line v = - 3u,

below by the line v = -3/ (1 + t), and on the right by the curve v
u (logu +log (1 + t) - 3). It is clear that K is a convex subset of JR^2.