1547671870-The_Ricci_Flow__Chow

2. SINGULARITY MODELS 235

An Einstein solution with R > 0 is Type I; an Einstein solution with
R < 0 is Type III; and a nonflat Einstein solution with R = 0 is Type Ilb.
Although Type Ila singularities are strongly conjectured to exist, it is be-
lieved (but not proved) that they will occur only for very special initial data.
The degenerate neckpinch discussed heuristically in Section 6 of Chapter 2
is simplest example in which a Type Ila singularity is expected to form.

REMARK 8.2. Perelman's recent work [105, 106, 107] does not explic-
itly distinguish between Type I and Type Ila singularities. Nonetheless, the
concepts of rapidly-forming and slowly-forming singularities are important
in so many aspects of the analysis of geometric evolution equations that we
will endeavor to help the reader be aware of the distinctions between these
singularity types for the Ricci flow.

2. Singularity models

Singularity models are complete nonflat solutions which occur as lim-

its of dilations about singularities. Any such solution exists for an infinite

time interval. Each type of singular solution (I, Ila, Ilb, and III) has its own

singularity model.

For insight, we turn again to the Einstein metrics. Notice that an Ein-

stein solution with R < 0 is an immortal solution: a solution that exists

on the future time interval [O, oo). Indeed, we may extend such a solution

to its maximal time interval of existence via the relation

9(t) = (1-

2

~(0)t) 9(0),

An Einstein solution with R > 0 on a time interval [O, T) with T < oo may

be uniquely extended backward in time to an ancient solution: a solution

that exists on the past time interval ( -oo, T). Here

9(t)= (1-~)9(0), t E (-oo, T).

Finally, an Einstein solution with R = 0 (namely, a Ricci-flat solution) may

be uniquely extended to an eternal solution: a solution that exists for all

time ( -oo, oo). Clearly, a Ricci-flat metric is static:

9(t)= 9(0) t E ( -oo, +oo).

Singularity models will turn out to b e solutions of one of these three types

• ancient, eternal, or immortal. (Recall that we saw other examples of
  ancient, eternal, and immortal solutions in Chapter 2.)

DEFINITION 8.3. Let (M~, 900 (t)) be a limit solution of the Ricci flow.

• One says (M~, 900 (t)) is an ancient Type I singularity model
  if it exists on a time interval (-oo, w) containing t = 0 and satisfies