1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

24 1. RICCI SOLITONS

Now we derive the asymptotic behavior of wand r. We have

d x

dY (log w + log Y) = y ( X _ a) ,

which by Lemma 1.33 approaches zero as Y-+ 0. Hence, wY approaches

some positive constant C as Y-+ 0. Next, note that

dr dY

w Jn(n -1) (X2 - aX)

shows that r is monotone increasing as Y decreases to zero, because X -a =

a(nx - y)/y is negative along this trajectory. Furthermore, the limiting

behavior of X and w implies that there is a constant c such that

dr w C

--= > ----~----

dY )n(n - l)X (a - X) v'n - 1(1 + c)Y^3

for Y sufficiently close to zero. We can similarly find a constant c^1 such that

dr 2(1 + c')C

--<----

dY y'n-1Y3

for Y sufficiently close to zero. Thus, we see by integration that r is O(Y-^2 )

as Y approaches zero. In particular, r -+ +oo. Therefore, the metric is '

complete. This proves the following.

THEOREM 1.35 (Existence of Bryant soliton). There exists a complete,

rotationally symmetric steady gradient soliton on JR.n+l which is unique up

to homothety.

REMARK 1.36. Since w = 0 (Y-^1 ) and r = O(Y-^2 ), we have w (r) =

0 (r^112 ). This tells us the Bryant soliton is like a paraboloid. Throughout

this section, by x = 0 (yP) we mean there exists a positive constant C such

that c-^1 yP :::; x :::; CyP. This is an abuse of notation since usually x = 0 (yP)

means lxl :::; CyP for some C.

4.4. Geometric properties of Bryant solitons. It is interesting to
contrast the asymptotic behavior of the curvature for these metrics with that
of the cigar metric. Recall from Chapter 2 of Volume One that the Gauss
curvature of the cigar falls off exponentially as a function of the distance
to the origin; in fact, since w(r) is asymptotically constant as r-+ oo, the
cigar is asymptotic to a cylinder. On the other hand, in a higher-dimensional
warped product, the n-dimensional volume of the sphere at distance r from
the origin is w(rrvol(Sn), where vol(Sn) is the volume of the standard n-
sphere. Since w(r) = O(r^112 ), we see that the sphere volume is unbounded.
From the limits in Lemma 1.33, we obtain the asymptotics of the deriva-
tives of w as functions of r:

w' = x = \/'n -1 X/Y = O(Y) = O(r-^112 )