1547671870-The_Ricci_Flow__Chow

5. THE NECKPINCH 41

REMARK 2.17. In the special case of a reflection-symmetric metric on
sn+l with a single neck at x = 0 and two bumps, the solution exhibits what
we call 'single point pinching'; to wit, the neckpinch occurs only on the
totally-geodesic hypersurface { 0} x sn' unless the diameter of the solution
(sn+^1 ,g(t)) becomes infinite as the singularity time is approached. (The
latter alternative is not expected to occur.)

We shall justify this remark in Subsection 5.6.

5.1. How the solution evolves. The first task in studying neck-
pinches is to compute basic geometric quantities related to the metric (2.42).
One begins by observing that g will solve the Ricci fl.ow if and only if c.p and
'ljJ evolve by

(2.46)

and

(2.47)

1/Jss

'Pt= n~c.p

1-1/J;

1/Jt = 1/Jss - ( n - 1) 'ljJ

respectively. In order that g ( t) extend to a smooth solution of the Ricci

fl.ow on sn+l, it suffices to impose the boundary conditions

(2.48) lim 1/Js = =fl.

X->±1

Indeed, if (2.48) is satisfied by the initial data, then the resulting solution

will satisfy the hypotheses of Lemma 2.10 for all times t > 0 that it exists.

REMARK 2.18. The partial derivatives Os and Ot do not commute, but

instead satisfy

[Ot, Os] = - n 1/J;s Os·

REMARK 2.19. Equation (2.46) will effectively disappear in what follows,

because the evolution of c.p is controlled by the quantity 1/Jss/1/J.

The Riemann curvature tensor of (2.43) is determined by the sectional

curvatures

(2.49) Ko= -1/Jss

'ljJ

of the n 2-planes perpendicular to the spheres { x} x sn' and the sectional

curvatures

(2.50) K 1 -- 1 -1jJ2 1/J;

of the G) = n(n - 1)/2 2-planes tangential to these spheres. The Ricci

tensor of g is thus

(2.51) Re= (nKo) ds^2 +(Ko+ (n - 1) K1) gcan,