1547671870-The_Ricci_Flow__Chow

38 2. SPECIAL AND LIMIT SOLUTIONS

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FIGURE 2. A neckpinch forming

5. The neckpinch
   The shrinking sphere we considered in Subsection 3.1 is the simplest
   example of a finite-time singularity of the Ricci fl.ow. In this section, we
   consider what is its next simplest and arguably its most important singu-
   larity, the 'neckpinch'.
   One says a solution (Mn, g ( t)) of the Ricci fl.ow encounters a local

singularity at T < oo if there exists a proper compact subset K c Mn

such that

but

sup IRml = oo

Kx [O,T)

sup IRml < oo.

(Mn \K) x [O,T)

This type of singularity formation is also called pinching behavior in

the literature. The first rigorous examples of pinching behavior for the

Ricci fl.ow were constructed by Miles Simon [119] on noncompact warped

products lRx f 5n. In these examples, a supersolution of the Ricci fl.ow PDE

is used as an upper barrier to force a singularity occur on a compact

subset in finite time. Another family of examples was constructed in [40].

Here, the metric is a complete U (n)-invariant shrinking gradient Kahler-

Ricci soliton on the holomorphic line bundle L - k over (::JP'n-l with twisting

number k E {1, ... , n - 1 }. Ast/ T, the CIP'n-l which constitutes the zero-

section of the bundle pinches off, while the metric remains nonsingular and

indeed converges to a Kahler cone on the set (Cn\ {O} )/Zk which constitutes

the rest of the bundle. Both of these families of examples live on noncompact

manifolds.