1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

86 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS

2.1.2. Examples.

Now we consider some examples.

EXAMPLE 19.16 (Cylinders have AVR = 0). If (pn-^1 ,k) is a closed

Riemannian manifold with Re :'.: 0, then the product (pn-l, k) x ffi. has

AVR = 0. Indeed, if (x, s) E pn-l xffi., then B ((x, s), r) C P x [s - r, s + r],
so that
VolB((x,s),r) :::;2rVol(P,k).

Although the above example is simple, in the next chapter we shall find
it useful in dimension 3.

EXAMPLE 19.17. For the Bryant soliton (Mn,9Bry) we have for any
pEM

(1) limr-+oo ~~~!~;;) E (O,oo) andAVR=Oand

(2) limd(x,p)-+oo R (x) · d (x,p) E (0, oo) and ASCR = oo.
Although in the next example the metric is incomplete near the vertex,
it is instructive to consider its geometry at infinity.

EXAMPLE 19.18 (The geometry at infinity of a Riemannian cone). Let

(pn-l, k) be a closed Riemannian manifold with Re :'.: 0 and consider the

Riemannian cone (GP, h), where

GP ~ ([o, 00) x pn-l) I rv

and {O} x {x} ,.._, {O} x {y} for all x,y E P and

We then have

(19.9)

and

h = dr^2 + r^2 k.

ASCR(h) < oo

AVR (h) = Vol (P, k) > 0.

nwn

EXERCISE 19.19 (The curvature of a Riemannian cone). To verify that

ASCR (h) < oo in the above example, compute the curvature of a Riemann-

ian cone (GP, dr^2 + r^2 k). HINT: Use (20.55) below.

Let (Mn, g, 0) be a pointed complete Riemannian manifold. The condi-

tion of the asymptotic cone^6 being top dimensional is related to ASCR (g) <

oo. In particular, suppose that the family {(Mn, ds-2 9 , 0)} sE[l,oo) converges
as s --+ oo in the pointed Gromov-Hausdorff distance to its asymptotic cone

(AX, dAx, 0). Furthermore, suppose that AX - {O} has a G^2 n-manifold

structure such that {(Mn, s-^2 g, 0)} sE[l,oo) converges ass--+ oo in the G^2

Cheeger-Gromov sense to a G^2 Riemannian metric g 00 on AX - {O}. Then

(^6) See subsection 1.3 of Appendix H.