1547671870-The_Ricci_Flow__Chow

114 5. THE RICCI FLOW ON SURFACES

COROLLARY 5.14. Under the normalized Ricci flow on a compact sur-
face, there exists a constant C such that

A consequence of this when r :::; 0 is that the metrics g ( t) are uniformly
equivalent for as long as a solution exists.

PROPOSITION 5.15. Let (M^2 , g (t)) be a solution of the normalized Ricci

flow on a compact surface with r :S 0. Then there exists a constant C 2: 1

depending only on the initial metric g (0) such that for as long as the solution

exists,

1

Cg(O) :S g(t) :SC g(O).

PROOF. It follows from (5.8) and (5.10) that

:tg = (r - R)g = (b,.J)g = (%/-rf) g.

Integrating this equation with respect to time implies

g(x, t) =exp [f(x, t) - f(x, 0) - r lot f(x, T) dT] g (x, 0).

Applying the corollary, we conclude that for any fixed vector V,

log ( g (V, V)l(x,t)) < C (ert + 1).

g (V, V) I (x,O) -

D

Now define

(5.11) H ~ R - r +IV' fl^2.

We expect H to satisfy a nice evolution equation, both because f itself does

and because H + r f = F - r is constant in space on Ricci gradient solitons.

PROPOSITION 5.16. On a solution (M^2 ,g(t)) of the normalized Ricci

flow on a compact surface, the quantity H defined in { 5.11) evolves by

(5.12)

where M is the tensor defined in ( 5. 9).

PROOF. Using (5.8), we rewrite equation (5.3) in the form

(5.13)

8

ot (R - r) = b,.R + R(R - r) = b,. (R - r) + (b,.!)^2 + r (R - r).