1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. CONSTRUCTING THE BRYANT STEADY SOLITON 23

for some c between a and b. Letting b-+ +oo gives g'(c)/h'(c) = g(a)/h(a).

Substituting in from (1.56) gives

:z (x~~-a~) +Jn~ - X ls=c =

2

:z ls=a.

Multiplying through by X^2 - aX (evaluated at c) and taking a -+ +oo

(which forces c-+ +oo) give

s-++oo lim y~ +a= 0.

However, to get zero on the right-hand side, we need to first know that

X/Y^2 is bounded as s -+ +oo.

We know that X/Y^2 ;:::: 0. To obtain an upper bound, we use the differ-

ential equation

_!,__ ds ( y2 X ) = a - y2 X (X^2 - 2aX + 1).

Since the trajectory approaches the origin in the XY-plane exponentially,

there exists a k > 0 such that

• ds d ( -y2 X ) < - ( y2 -X - a ) ( e-ks - 1 ) + ae-ks.

Comparing with the solution of the ODE

du/ds = u(e-ks - 1) + ae-ks

shows that X/Y^2 is bounded above. In particular, for s;:::: so

:Z (s):::; u (s) = e-(te-ks+s) (1: ete-ks+sae-k^8 ds + e-t :Z (so)).

The second limit, which gives the next term in the power series expansion
of X in terms of Y near the origin, follows by applying the same arguments
to the equations

:s ( X ~~y

2

) = (-1-3X^2 + 4aX) ( X ~~Y

2

)

- ; 2 (X^2 - 2aX)

and

fs (X -aY

2

) = (X -aY

2

) ( -1 )

-4.y4 ds y4 4 (X^2 - aX)

a X^3

• 2Y^2 + 4Y^4 (X^2 - aX).
  D

EXERCISE 1.34. Verify the second limit.