1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

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  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.
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