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10. DIFFERENTIAL HARNACK ESTIMATES OF LYH TYPE 143

The uniform upper bound for the scalar curvature implies a uniform
upper bound for the diameter.

COROLLARY 5.52. For any solution (M^2 ,g(t)) of the normalized Ricci
flow on a closed surface with R (., 0) > 0, there exists a constant C > 0
depending only on go such that

diam(M^2 ,g(t))::::; C.

PROOF. Suppose there are N points p1, ... ,PN E M^2 such that

dist g(t) (Pi, Pj) 2:

2

7r

)Rmax (t)

for all 1 :Si # j :::; N. It follows from Klingenberg's Lemma that

·. (M^2 (t)) > 1T

lllj ,g - )Rmax (t) /2'

hence that the balls Bg(t) (Pi,7r/)Rmax(t)/2) are embedded and pairwise

disjoint. The area comparison estimate then implies that there exists c > 0
such that

Area(M^2 ,g(t)) 2: tAreaBg(t) (Pi, 1T ) 2: N c ()

i=l J Rmax (t) /2 Rmax t

Hence by Proposition 5.51, there exists C > 0 such that

N::::; Rmax (t) Area (M^2 , g (t)) :::; C ·Area (M^2 , go).

c

0

10. Differential Harnack estimates of LYH type

In the context of parabolic differential equations, a classical Harnack

inequality is an a priori lower bound for a positive solution of a parabolic

equation at some point and time in terms of that solution at another point

and an earlier time. In their important paper [ 92 ], Li and Yau obtained new

space-time gradient estimates for solutions of parabolic equations. Their es-

timates are differential inequalities which substantially generalize classical

Harnack estimates. Moreover, their method relies primarily on the maxi-

mum principle, hence is robust enough to work in the context of Riemannian

geometry. Because of later advances made by Hamilton in the ideas Li and

Yau pioneered, such estimates are today called differential Harnack es-

timates of LYH type.

We plan to discuss differential Harnack estimates for the Ricci flow in

n dimensions in a chapter of the successor to this volume. Here, we de-

rive certain differential Harnack inequalities that let us estimate the scalar

curvature function on a surface evolving by the normalized Ricci flow.