1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

28. 
    
    
    
    1. RICCI SOLITONS
    
    
    
    

interested in solutions in the half-space where. w is positive. There is a 1-

parameter family of these trajectories in the unstable manifold, and they are

tangent to the vector (1, 0, 0) as they exit from P. (This vector is associated

to the positive eigenvalue closest to zero.) Among these is the hyperbolic

metric, for which f' = 0, y = nx, and the warping function is

w = Vnf>.sinh ( ~r), y = ncosh ( ~r).

5.2. Analysis of a 1-parameter family of trajectories. For the

rest of this section, we will consider only the 1-parameter family of trajec-
tories that lie in the quadrant of the unstable manifold of P, between the

hyperbolic metric and the steady soliton, i.e., those for which w and y - nx

are positive for small values of r. Among these is the flat Gaussian soliton

on ffi.n+l described in Example 1.9; the corresponding solution of (1.59) is

w = r, x = 1, y = n + .Ar^2 •
In order to show that the metrics corresponding to our family of tra-
jectories are complete and to study their curvatures, we again introduce
rescaled coordinates
w
W~-,
y

x

X~Vr/,-,

y

y ~ yfn(n-l)_

y

We introduce a new independent variable s such that ds = y dt, whereupon

syste~ (1.59) becomes ·

dW = W(X2 - .AW2)

ds '

dX

(1.60) ds = X^3 -X + aY^2 + .A(Vr/,-X)W^2 , a~ 1/Vr/,,

dY 2 2

ds =Y(X -aX-.AW ).

In these coordinates, P = (0, a, Vl - a^2 ), and the hyperbolic trajectory has
X identically equal to a, approaching the critical point H ~ (1/..;n5:.., a, 0)
as r '......-t oo. The trajectories we are considering lie in the unstable manifold

of P, and for small r, in the region where X < a and W > 0. The flat

trajectory satisfies Y = .;n=l X and X^2 - aX + .AW^2 = o..

LEMMA 1.39. These trajectories remain in the region defined by

0 < X <a, 0 < y ~ Vl-X^2 , 0 < w ~ 1;..;n>:...

PROOF. Along the plane where X = a,

dX/ds = a^3 - a+ aY^2 +.A( Vi/,-a)W^2.

Let Q stand for the quantity on the right, which must be negative for r

small. The following equation shows that Q remains negative if X < a:

dQ/ds = aXY^2 (X - a)+ .A(Vr/,-a)(X^2 - a^2 )W^2 - .AQW^2.