1547671870-The_Ricci_Flow__Chow

118 5. THE RICCI FLOW ON SURFACES

Since R (x, t) :::; p (t) :::; 0, this is possible only if

IRc-~gl2 = 0

at ( x, t). Tracing then implies that

Rmin ( t) = R ( X, t) = p ( t) ,

hence that R = r is constant in space. Substituting back into (5.16), we
conclude that

Rc(·,t) = p(t)g(·,t)
n
holds identically, hence that g (t) is Einstein. D

4.2. Surface solitons and the Kazdan-Warner identity. Now we
prove the main assertion of this section. Since the argument uses a partic-
ular formulation of the Kazdan- Warner identity [82], we follow it with a
discussion of that result.

PROPOSITION 5.21. If (M^2 ,g (t)) is a self-similar solution of the nor-

malized Ricci flow on a Riemannian surface, then g(t) = g(O) is a metric of

constant curvature.

PROOF. By Proposition 5.20, we may assume that r > 0. By passing to

a cover space if necessary, we may thus assume that M^2 is diffeomorphic to

52. Contracting the Ricci soliton equation (5.6) by Rg-^1 yields

2R (r - R) = 2Rdiv X,

and hence

• f (R-r)^2 dA= f R(r- R)dA= f RdivXdA.

Js2 Js2 ls2

Since X is a conformal Killing vector field, integrating by parts and applying

the Kazdan-Warner identity (5.18) implies that

f (R - r)^2 dA = f (\7 R, X) dA = 0.

Js2 Js2

Hence R = r, and the proposition is proved. D

We begin our discussion of the Kazdan- Warner identity by stating its

usual formulation. (See for example page 195 of [113].) Let g be an arbitrary

metric on a topological 52. We shall denote the standard round metric on

52 by g and will use bars to indicate geometric quantities associated to g.

Let t.p be a first eigenfunction of the rough Laplacian ~ on 52. (In other

words, cp is a spherical harmonic.) By the Uniformization Theorem, one may

write g = e^2 ug. Then one can write the standard Kazdan- Warner identity

in the form

(5.17)