1547671870-The_Ricci_Flow__Chow

5. THE NECKPINCH 49

where U is the (2,0) tensor given by

U =(Ki - Ko) [(n - 1) ds^2 + ~^2 gcan].

To verify (2.56) at any given point (x, P) E (-1, 1) x sn, we adopt the con-

vention that Roman indices belong to {O, ... n} while Greek indices belong

to {l, ... , n}.

We choose coordinates {yi, ... , yn} near P on sn such that g = gcan has
components flaf3 = l5af3 at P. Then we set y^0 = s so that {y^0 , y1, ... , yn}

is a coordinate system near ( x, P) on ( -1, 1) x sn. The only nonvan-

ishing components of the metric g in these coordinates are goo = 1 and

g°'°' = ~^2 ?Jaa· Moreover, all components of the Riemann tensor vanish in
these coordinates except Raooa = ~^2 Ko and Raf3f3a = ~^4 Ki (o:-=/= /3). Sim-
ilarly, all components of the Ricci tensor vanish except Roo = nKo and
Raa = ~^2 [Ko+ (n - 1) Ki]. The evolution equation (2.56) now follows
from formula (6.7). We have established (2.56) on the punctured sphere

(-1, 1) x sn; by continuity it remains valid at the poles {±1} x sn.

To apply Proposition 2.33, let Nt denote the topological (n + 1)-ball

Nt = {(x, P): x 2: x* (t), PE sn}

endowed with the metric g (t). Observe that the sectional curvatures Ko
and Ki are strictly positive on

&Nt = {x* (t)} x sn,

because~ has a local maximum at x* (t) and ~s has a simple zero there by

Lemma 2.31. So Re > 0 on &Nt. If~(-, 0) is strictly convex for all x 2: x* (0),

then Re(·, 0) > 0 on Nt. So if Re ever acquires a zero eigenvalue, it must

do so at some point p E intNt and time t E (0, T). If Re (V, V)l(p,t) = 0

for some vector V E TpNt, then (U *Re) (V, V) = 0, because U and Re

commute. Hence Lemma 2.33 implies that Re 2: 0 on Nt for as long as g (t)

exists. The lemma follows immediately. 0

Finally, we prove that when rt > 0 is chosen sufficiently small, the quan-

tity

b =. ~ ~T/ =. 1<p ,2- 71 IK i -KI 0

remains bounded in a neighborhood B of the pole. Because the exponent rt

breaks scale invariance, b may be regarded as a pinching inequality for the

curvatures on the polar cap. The bound on b thus lets us apply a parabolic

dilation argument that shows that singularity formation on a polar cap under

our hypotheses would lead to a contradiction. (We shall study parabolic

dilations in greater generality in Chapter 8.)

LEMMA 2.35. Let g (t) be as in Proposition 2.30. If D > 0, no singularity

occurs on the polar cap