1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. THE THREE CASES OF NONSINGULAR SOLUTIONS 205

Each higher time derivative of r (t) is an integral of a polynomial expression of Rm

and its space derivatives. In view of this, the proof of the compactness theorem in

[139] goes through without much change. In particular, in the limit one obtains a

solution to (32.5). D

EXERCISE 32.6.

(1) Show that for any n ;:::: 3 there exists a sequence of closed pointed Riemann-

ian manifolds { (Nn, hi , Xi)} with average scalar curvatures rh; ---+ +oo and

which converge to a complete noncompact pointed Riemannian manifold

(N-::O, h 00 (t), x 00 ) with bounded curvature. Note that this is not possible

for n = 2 by the Gauss-Bonnet formula.

(2) Show that for any n;:::: 3 there exists a sequence of closed pointed Riemann-

ian manifolds { (Nn, hi, xi)} with uniformly bounded curvature which con-

verges to a complete noncompact pointed Riemannian manifold (N-::O, h 00 ,

x 00 ) and for which the sequence {rhJ does not converge; in particular,

there exist two subsequences for which the average scalar curvatures con-

verge to two different numbers.

HINT: (1) Let part of (Nn, hi) be isometric to 5n-^1 (t) x (O,i^3 ), which drifts

off to infinity relative to Xi as i ---+ oo.

(2) Consider the set of points in JRn+l of distance 1 from the disk Dn(i) x {O}.

(Pancake) For i odd, let (Nn, hi) be a smoothing of this C^1 hypersurface.

( Umbrella pole and base) For i even, let (Nn, hi) be the same attached to a

smoothing of sn-^1 ( 1) x [1, in] capped off at the top.

REMARK 32.7. Corollary 3. 18 in Part I, which was intended to be a consequence
of Hamilton's compactness theorem, is incorrect as stated. We would like to thank
John Lott and Peter Topping for pointing this out to us. For correct statements,

see Corollaries E.2 and E.4 in [161] and see [418].

3.3. Limits as ti---+ oo via the compactness theorem.

Let (M^3 , g (t)), t E [O, oo ), be a noncollapsed nonsingular solution to the NRF

flow on a closed 3-manifold. Then there exists 6 > 0 such that for any ti ---+ oo

there exist points Xi E M such that

(32.6) inj g(t;) (xi) 2: 6.

To such a sequence { (Xi, ti)}, consider the corresponding sequence of pointed so lu-
tions (obtained by translating backward in time)

(32.7)

By Theorem 32.5, there exists a subsequence which converges to a complete pointed
solution

(32.8) (M~,g 00 (t) ,x 00 ), t E (-00,00),

of equation (32.5), where the limit solution has uniformly bounded curvature and

finite volume Vol (g 00 (t)) :::; Vol (g (0)). We call (M 00 , g 00 (t), x 00 ) a limit solution

corresponding to {(xi, ti)}.