106 5. THE RICCI FLOW ON SURFACES
attempt to present the most efficient proofs known. The organization of the
chapter is as follows. In Sections 1 through 4, we study the evolution of
a metric on an arbitrary surface and derive certain a priori estimates that
apply to all 2-dimensional solutions (M^2 , g (t)) of the Ricci fl.ow. Then we
establish Theorem 5.1 by considering the three natural cases. In Section 5,
we prove exponential convergence to a metric of constant negative curvature
in the case that x (M^2 ) < 0. In Section 6, we prove convergence to a fl.at
metric on surfaces satisfying x (M^2 ) = 0. In the remainder of the chapter,
we treat the far more difficult case that x (M^2 ) > 0, following the strategy
outlined in Section 7.REMARK 5.2. The Ricci fl.ow still has not provided a complete proof of
the Uniformization Theorem if x (M^2 ) > 0. As we shall see in Section 7, the
proof of Theorem 5.1 for this case uses the Kazdan-Warner identity to show
that the only Ricci solitons on 52 are the metrics of constant curvature. The
proof of the Kazdan-Warner identity requires the Uniformization Theorem.
(In [60], Hamilton gives another proof that the only solitons on 52 are
trivial. But this proof uses the fact that 52 \ { point } is conformal to IR^2 ,
hence also requires U niformization.)1. The effect of a conformal change of metric
Let us recall how the curvature, Laplacian, and volume element of a Rie-
mannian manifold (Mn, g) transform under a conformal change of metric.
To study the curvature, we shall use moving frames. Let {ei}~ 1 be a
local orthonormal frame field: a local orthonormal basis for T Mn in an open
set UC Mn. The dual orthonormal basis of T* Mn (induced coframe field)
is denoted by { wi} ~=land defined by wi (ej) = 8j for all i,j = 1, ... , n. The
metric is then given by
n
g = Lwi®wi.
i = l
The connection 1-forms wf E !1^1 (U) are the components of the Levi-Civita
connection with respect to { ei} ~ 1 and are defined by\7xei = wf (X) ej,
for all i, j = 1, ... , n and all vector fields X E c= (T Mnlu). They are
antisymmetric: wf = -wj. We write the first and second structure equations
of Cartan in the form(Note the Einstein summation convention.)