1547671870-The_Ricci_Flow__Chow

116 5. THE RICCI FLOW ON SURFACES

for a short time. We call these Bernstein-Banda-Shi (BBS) derivative
estimates. They originate in Bernstein's ideas [16, 17, 18] for proving
gradient bounds via the maximum principle, and were derived for the Ricci
flow in [9] and [117, 118]. (We begin our analysis of BBS estimates in
Sections 5, 6, and 9. We discuss their general n -dimensional formulation
and provide further references in Chapter 7.)
STEP 2. Short-time existence results (Theorem 3.13 and Corollary 7.7)
tell us that the lifetime of a maximal solution (Mn, g (t)) is bounded below
by
c
maxMn !Rm [go] 190 '

where c is a universal constant depending only on n. (See also the doubling-
time estimate of Lemma 5.45.)
STEP 3. Long-time existence results (Theorem 6.45) then imply that
the flow cannot be extended past T < oo only if

lim ( sup !Rm (x, t)i) = oo.

t/T xEMn

Because of these facts, the bounds in Proposition 5.18 allow us immedi-

ately to apply Corollary 7.2 to get the following long-time existence result

for solutions on surfaces.

PROPOSITION 5.19. If (M^2 , go) is a closed Riemannian surface, a unique

solution g(t) of the normalized Ricci flow exists for all time and satisfies

g (0) = 90·

In Sections 5 and 6, we will derive bounds for all derivatives of the

curvature on surfaces of nonpositive Euler characteristic, thus enabling a

direct proof of long-time existence in those cases. In Section 9, we apply the

BBS method to bound the gradient of the scalar curvature on a surface of

positive Euler characteristic. Finally, in Chapters 6 and 7, we will prove more

general results that yield bounds on all higher derivatives of the curvature

and establish maximal-time existence (under appropriate hypotheses) on

compact manifolds in all dimensions.

4. Uniqueness of Ricci solitons

The main objective of this section is to prove that the only gradient Ricci

solitons on surfaces are the metrics of constant curvature. This fact will be

important when we consider the long-time behavior of the normalized Ricci

flow on a surface of positive Euler characteristic. We start with a result that

holds in all dimensions.

4.1. Uniqueness of steady and expanding solitons. If (Mn,g) is

a compact Riemannian manifold, let us denote its volume by

v ~ r dμ

}Mn