- THE K ING-ROSENAU SOLUTION IN THE VARIO US COORDINATES 75
3.2. A quantity that is constant on the King-Rosenau solution.
Now we discuss some properties of the King- Rosenau solution. Given the coor-
dinates (x^1 , x^2 ) = (1/J, B) on 52 - {N, S}, we have that t he only nonzero components
of t he Christoffel symbols rfJ of g 52 are
(29.18) rl 22 - - sin -2-. 27/J
Hence, for any radial function f = f ( 'ljJ) on 52 we have
f:::.s2f = gij ( ():i
2
fxJ - r~j ()~k) = f.1/n/; - tan 'ljJ f ..p.Thus the pressure function vKR of the King- Rosenau solution satisfies the equation
(29.19) F ~ 6. 5 2vKR + 6vKR = 4(3 + 12 0:.
This is significant since quantities which are co nstant in space on the desired special
solutions often are good quantities to which to apply the maximum principle for
general solut ions.
3.3. Backward limits of the King-Rosenau solution.
We describe how the cigar soliton is a backward Cheeger- Gromov limit of the
King- Rosenau solution. For simplicity, let μ = 1 and let N = (0, 0) in Euclidean
coordinates be our basepoint. Then by Lemma 29.3 we have that
uKR(r, e, t) = - ( ~ csch( 4t) + r^2 coth( 4t) + r
2
4
csch( 4t) )-1Let ti ~ -oo and let^1
gfR(t) ~ ufR(t)geuc,where ufR(x, y , t) ~ K[uKR(Kix, Kiy, t +ti) and Kl~ (uKR)-^1 (0, 0, ti).
Then K[ = -~ csch(4ti) =a (ti) and
ufR(r, B, t) = (a (ti)-l a (t +ti) + r^2 b (t +ti)+ r^4 a (ti) a (t +ti) )-l,
where b (t) = -coth(4t). Thus
(29.20) uKR(r B t) = lim uKR(r B t) = (e^4 t + r^2 )-
1
00 ) '. i-too. i , ' 'or , equivalently,
(29.21) VKR(r 00 ' e ' t) =. (uKR) 00 -^1 (r ) e ' t) = e^4 t + r^2.
This says that g~R ~ u~Rgeuc = gcig· We have proven the following:
LEMMA 29 .4 (Cheeger- Gromov convergence to a cigar at the poles). Let (5^2 ,
gKR) be the King-Rosenau solution. For any sequence ti ~ -oo, let <Pi : 52 ~ 52
be th e conformal diffeomorphisms of 52 characterized by <Pi(N) = N, <Pi(S) = S,
and <PigKR(N, ti) = gs2(N) for each i. Then (5^2 , <PigKR(t +ti), N) converges in
the C^00 pointed Cheeger- Gromov sense to a cigar soliton solution. The same result
holds true for N replaced by S.
(^1) We use a "tilde" instead of a "bar" in our notation for 9i and Ui to remind us that the
sequence invo lves a conformal diffeomorphism change.