124 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONSSince 1 = c;, we have that ~g\ is independent oft. On the other hand, by pulling
back by a conformal diffeomorphism, we can rescale the metric and coordinates bydefining for any A > 0
ii>.(x, y , t) ~ ,>_-^2 v(.>-x, .>-y, t)= .>- 2 A ( x2 + y2) 2 + B ( x2 + y2) + .>--2 C.
So, without loss of generality, we may assume that A (t) = C (t) for all t. We then
obtain
ii(x, y , t) =A (x^2 + y^2 )2
+ B (x^2 + y^2 ) +A,
A'= 4BA, B' = 16A^2.
This is (29.14) and (29.15); so we have obtained the King- Rosenau solution.14. The evolution equation for Q
In this section we consider the evolution equation for Q, which is the plane
version of equation (29.140) for Q. We calculate this directly for the sake of com-
parison; some aspects of the calculation are simpler because the background space
is flat. The reader may skip this section if he or she likes since the discussion is not
necessary for the proof of the main theorem of this chapter.
Let a be the 3-tensor defined by (29.171) and let TF (/3) denote the trace-free
part of a tensor f3 of degree 3 or 4 as defined in (29.131) or (29.156), respectively.LEMMA 29.48. The quantity Q =ii lal^2 on ffi.^2 defined in (29.172) evolves by(29.196) ~~ = ii6eucCJ - 4RQ - 2 (1TF(ii\i'a+2dii@ a)l^2 +I ~z - dii 81 ,1 al
2
)
::; ii6eucQ,w h ere z :::;= • n 2 A v UeucV -- 2 ueuc^1 A^2 vg1R2 - an d (-v 81,1 -) a ij = L..'\'e=l^2 - -veaeij.
We now sketch a proof of the lemma. Taking three partial derivatives of
(29.170), we obtain(29.197)= ii6 eucVijk + Vijk 6 eucii + V[i6euciijk] + V[jk6euciii]
- 2iieijkiie - 2iie[jkiii]e,
where A[ijk] ~ Aijk + Ajki + Akij for a 3-tensor A. Thus
(29.198)
8
Bt 1-Vijk^12 = Vueuc -A 1-Vijk^12 - 2-1-V VJ!ijk^12 + 2A t....leucV -1-Vijk^12
+ 6iiijk (iii6eucVjk + Vjk6 euciii) - 4iiijkVeijkiie - 12iiijkVejkV£i·Here and throughout, we include the indices in our notation for the norm of a
tensor; for example, liiijkl^2 = l8^3 vl
2
.
Tracing (29.197) yields
8