- UNIFORM UPPER BOUNDS FOR R AND IV RI
PROOF. By (5.10) and (5.2),Hence
:t JM 2 f dA = JM 2 ((!:lf + rf) + f (r - R)) dA
= J ((!:lf + r !) - f !:lf) dA
J.,12= JM 2 (r f +IV f\
2
) dA.!!__ ( e-rt J f dA) = e-rt r IV l 12 dA.
dt J.,12 }M2137But by Corollary 5.14, we have f M 2 Ill dA :S Cert for some C < oo, so that
integrating the equation above with respect to time yields
{Te-rt r IV' 112 dA dt = [e-rt J f dA] t=T :S 2C.
Jo }M^2 M^2 t=O
D
PROPOSITION 5.44. Let (M^2 , g ( t)) be a solution of the normalized Ricci
flow corresponding to an arbitrary initial metric with r > 0. Then there is
a constant C depending only on g (0) such that
N(g(t),s(t))::::; c.
PROOF. By (5.28), there is C > 0 depending only on g (0) such that-Ce-rt :S s :S 0. Hence (5.30) implies that
!!__N :S -s { (IV! 1^2 + s - r - (R - s) log (R - s)) dA
dt }M2:S ce-rt JM2IV112 dA + ce-rt INI.
The proposition follows by integrating this inequality with respect to time
and applying Lemma 5.43. D9. Uniform upper bounds for Rand IV' RI
In this section, we derive further estimates for a solution g ( t) on a surface
of positive Euler characteristic. In the first part, we obtain a key doubling-
time estimate, and derive upper and lower bounds for g ( t). In the second
part, we derive a BBS estimate for the gradient of the scalar curvature. In
the final part, we apply the entropy estimate to obtain uniform bounds for
the scalar curvature itself under the additional hypothesis that R (·, 0) > 0.