70 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS
In §2 we review the various standard coordinates in which we express the Ricci
flow equation on 52. Based on Cheeger- Gromov limits for times ti ---+ -oo, we
develop intuition to guide the proof.
In §3 we describe the King- Rosenau solution in natural coordinate systems on
the sphere, the plane, and the cylinder. We consider various backward limits of the
King- Rosenau so lution, which help our study of nonround ancient solutions on 52.
In §4 we prove a priori estimates for the pressure function v (reciprocal of the
conformal factor relative to the round metric) of an ancient solution.
In §5, we give two proofs that the backward limit of the scalar curvature is
equal to zero a.e. The first proof uses the evolution of the area form and the second
proof uses an energy monotonicity formula (a fine estimate).
In §6, by applying the a priori estimates in the previous sections, we show
that the backward limit of the pressure function corresponds to a fiat (possibly
incomplete) metric in a weak sense.
In §7, we discuss some basic properties of the isoperimetric constant of metrics
on 52.
In §8, using isoperimetric monotonicity, we prove that if one has infinite expan-
sion of the conformal factor backward in time, then the solution must be round.
In §9, by improving the result of a previous section, we show either that the
backward limit of the pressure function is zero (i.e., the metric has infinite expansion
backward in time) or that the b ackward limit corresponds to a complete fiat cylinder
metric. Some key tools are a concentration-compactness-type result, a monotonicity
formula for circular averages, and the classification of harmonic functions on IR^2 with
at most linear growth.
In §10 we prove that, for nonround solutions, we also obtain cigar backward
limits at the poles, where the locations of the poles are determined by the cylinder
limits.
In §11 we consider the key quantity Q, which depends on the third covariant
derivative of the pressure function.
In §12 we prove that Q is a subsolution of the heat equation.
In § 13 we show that if Q is identically zero, then the ancient solution is the
King- Rosenau solution.
In §14 we consider the quantity Q on the plane analogous to Q.
In §15 we prove that Q is identically zero.
In §16 we show that the quantity Q on 52 is the pull-back by stereographic
projection of the quantity Q on IR^2.
2. The Ricci flow equation on 52 and some intuition
In this section we review the spherical, Euclidean, cylindrical, and polar coor-
dinates on 52 minus 0, 1, or 2 points that we shall use to analyze ancient solutions
to the Ricci flow on 52. We introduce the backward limit of the pressure function
and we also discuss some intution which will guide our analysis.
2.1. The pressure function and its equation in various standard co-
ordinates.
Let 52 denote the unit 2-sphere centered at the origin in IR^3 , let N = (0, 0, 1)