1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. SINGULARITY MODELS AND K;-SOLUTIONS 81

and n such that if (Mn, g) is a complete Riemannian manifold, p E M, and

r E (0, oo) are such that JRmJ ::::; r-^2 in B (p, r) and Vol~~p,r) 2: r;,, then

inj (p) 2: Lor.

Combining the above two results with Hamilton's Cheeger-Gromov-type

compactness theorem, we have the following.

THEOREM 19.4 (Existence of finite time singularity models). Let g(t),

t E [O, T), be a solution to the Ricci flow on a closed manifold Mn with

T < oo. Suppose that we have a sequence of times ti/" T, points Pi EM,

radii r i E ( 0, oo), and a constant C < oo such that

(19.2) Ki ~ I Rm I (pi, ti) ---+ oo,

(19.3) I Rm I (x, t) ::::; CKi for all x E Bg(ti) (pi, ri) and t <ti,

(19.4)

Then there exists a subsequence of rescaled solutions

gi ( t) ~ Ki · g (ti + ;J

such that (Mn, gi(t),pi) converges to a complete ancient solution

(M~, goo(t),Poo)

in the C^00 Cheeger-Gromov sense. This limit solution has nonnegative

scalar curvature and is r;,-noncollapsed at all scales, where r;, > 0 is a con-

stant depending only on g ( 0) and T.

Recall that the limit solution (M~, g 00 (t)) is called a finite time singu-

larity model.

EXERCISE 19.5. Prove Theorem 19.4.

HINT: Compare with Corollary 3.18 and Corollary 3.29 in Part I. Note

that if the solution g (t) is r;,-noncollapsed below the scale p, then the solution

gi (t) is r;,-noncollapsed below the scale Ki^1 /^2 p, while Ki ---+ oo.

REMARK 19.6. There is a corresponding result for a complete solution on

a noncompact manifold with bounded curvature provided the initial metric

is r;,-noncollapsed below some scale p; see Theorem 8.26 in Part I.

Now recall the following important concept originally introduced by

Perelman in §11.1 of [152] (see also Definition 8.31 in Part I).

DEFINITION 19.7 (r;,-solution). Given a positive constant r;,, a complete
ancient solution (Mn,g(t)), t E (-oo,O], of the Ricci flow is called a r;,-
solution if it satisfies the following.

(i) For each t E (-oo, O] the metric g(t) is nonfiat with nonnegative

curvature operator and r;,-noncollapsed at all scales.