CHAPTER 5
The Ricci flow on surfaces
One of the triumphs of nineteenth-century mathematics was the Uni-
formization Theorem, which implies that every smooth surface admits an
essentially unique conformal metric of constant curvature. This result may
be interpreted as the statement that every 2-dimensional manifold admits a
canonical geometry. Looked at another way, it may be regarded as a classi-
fication of such manifolds into three families - those of constant positive,
zero, or negative curvature. In this sense, the Uniformization Theorem may
be regarded as a forerunner, at least in spirit, of current efforts to classify
the geometries of closed 3-manifolds.
In this chapter, we study the Ricci flow on closed 2-dimensional mani-
folds. We shall show that a solution of the normalized Ricci flow exists for all
time and converges to a constant-curvature metric conformal to the initial
metric. The existence of a constant curvature metric in each conformal class
is a classical fact which is equivalent to the Uniformization Theorem. So in
this sense, the Ricci flow may be regarded as a natural homotopy between
a given Riemannian metric and the canonical metric in its conformal class
whose existence is guaranteed by the Uniformization Theorem.
If (M^2 , g) is a compact Riemannian surface (to wit, a 2-dimensional
Riemannian manifold) we denote its average scalar curvature by
By the Gauss-Bonnet theorem, r is determined by the Euler characteristic
x (M^2 ) of the surface, hence is independent of the metric g. The objective
of this chapter is to prove the following result.THEOREM 5.1. If (M^2 , go) is a closed Riemannian surface, there exists
a unique solution g (t) of the normalized Ricci fiow
8
8tg = (r - R) g
g (0) =go.
The solution exists for all time. As t ---+ oo, the metrics g (t) converge
uniformly in any Ck -norm to a smooth metric g 00 of constant curvature.One of the objectives of this chapter is to introduce important tech-
niques. Therefore, in the interest of pedagogical clarity, we shall not always
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