- THE KING-ROSENAU SOLUTION IN THE VARIOUS COORDINATES 73
2.2. Intuition based on earlier results.
To gain intuition, we note the following earlier results of Hamilton on backward
Cheeger- Gromov limits.
THEOREM 29.2 (2-dimensional Type I and II ancient solutions).(1) If K ~ sup 5 2x(-oo,-l] JtJR < oo (i.e., g (t) is Type I), then g (t) must be
a round shrinking 2-sphere. In this case, a backward pointed limit is the
fiat plane.
(2) If K = oo (i.e., g (t) is Type II), then there exists a backward Cheeger-
Gromov limit which is the cigar soliton. By taking a second limit of points
tending to infinity in the cigar, we obtain a fiat cylinder. Since the limit
of a limit is a limit, we have that the fiat cylinder is a backward Cheeger-
Gromov limit of g(t).
Part (1) can be proved by using Hamilton's entropy monotonicity and part (2)
can be proved by using Hamilton's trace Harnack estimate. For expositions, see
Theorem 9.14 and Proposition 9.18, both in [77]. A priori, it may require rescaling
to obtain the limit in part (2). However, Proposition 29.38 below obtains cigar
soliton limits without needing to rescale. Note that the main theorem, Theorem
29 .1, says that if g (t) is Type II, then g (t) must actually be the King- Rosenau
solution.
One should make the distinction between Cheeger- Gromov limits, which use
t he pull-backs by diffeomorphisms, and pointwise limits. We shall use both types
of limits in the proof of the main theorem. Furthermore, we expect to h ave (and
in fact do have) the following correspondences:
(1) If a backward Cheeger- Gromov limit is the fl.at plane, then g(t) is a round
shrinking 2-sphere.
(2) If a backward Cheeger- Gromov limit is a flat cylinder or a cigar, then g(t)
is the King- Rosenau solut ion.3. The King- Rosenau solution in the various coordinates
In this section we characterize the King- Rosenau solution as the only solution,
up to a constant scale factor, having a specific rotationally symmetric form given
below. This characterization is used in the proof of the main theorem. We also make
some observations about the behavior of certain quantities on the King- Rosenau
solution.
3.1. Characterization of the King-Rosenau solution.
We shall use the superscript or subscript KR for quantities on the King-
Rosenau solution. A general method for finding explicit solutions of PDE, including
those which arise in geometry, is to find a suitable ansatz. The ansatz for the
King- Rosenau solution on 52 is gKR(t) = ~g52, where
(29.8) VKR ('ljJ, 8 , t) = 2a (t) + (3 (t) COS^2 'ljJ
and where a (t) and (3 (t) are positive functions of time. In terms of the induced
coordinates on the unit 2-sphere as a subset of IR^3 , we have
(29.9) vKR (x, y, z , t) = 2a (t) + (3 (t) (1 - z^2 ),
which is evidently smooth on 52.