1547671870-The_Ricci_Flow__Chow

62 2. SPECIAL AND LIMIT SOLUTIONS

estimates at (s, f) that

V - t - Vt -< c (-f}S f)t - p I - --n 'l/J - 2 ) V - (n - 1) (1 - V -^2 ) -'l/;2 V

(

f)s ') n - 1 v

::;; c 8t - p - - 2- 'l/; 2

=rn[~-f W' ds]

< ~ [en - n 2~ 1].

n-l v

• 
  
  
  
  • 2 - 'l/;2
  
  
  
  

Choose c3 < (n - 1)/[2n'l/Jmax(O)]. Then if 0 < c < min{c1,c2,c3}, the

consequence 'l/Jt < 0 of Lemma 2.27 implies the inequality 'Y..t - Vt < 0. This
contradicts (2.67), hence proves the result. D

6. The degenerate neckpinch

In contrast to the contents of Section 5, the discussion in this section is

heuristic and not fully rigorous. Nonetheless, degenerate neckpinches have

been rigorously demonstrated [7] for the mean curvature flow of a surface

in JR^3. Although an analogous result is still lacking for solutions of the
Ricci fl.ow, the mean curvature fl.ow is similar in so many respects that it is
strongly conjectured that degenerate neckpinches exist for the Ricci fl.ow.

6.1. An intuitive picture of degenerate neckpinches. Let us con-

sider how a degenerate neckpinch should arise. Imagine a family of rota-

tionally symmetric solutions

{(Sn,ga (t)): a E [O, 1]}

of the Ricci flow parameterized by the unit interval. When a = 0, let

the initial metric have the profile described in Subsection 5.5. This is a

symmetric dumbbell with two equally-sized hemispherical regions joined by

a thin neck. (See Figure 2.) We proved in Section 5 that there exist such

initial conditions which lead to a neckpinch singularity of the Ricci flow at

some time To < oo. On the other hand, when a = 1, let the initial metric be

a round metric on then-sphere. As we saw in Subsection 3.1, such a metric

remains round and shrinks to a point at some time T1 < oo.

Now suppose that for a close to 1, the initial metric 9a (0) is a lopsided

dumbbell where the neck is fat and short and one of the hemispheres is

smaller than the other. In this case, if a is close enough to 1, the smoothing

effects of the Ricci flow are conjectured to be strong enough to allow the

smaller hemisphere to pull through before the neck shrinks. (See Figure

5.) In this case, the metric should eventually acquire positive curvature

everywhere and shrink to a round point. (This is of course known if 9a (0)

is a metric of positive Ricci curvature on the 3-sphere; see Chapter 6.)