132 5. THE RICCI FLOW ON SURFACES
curvature allows us to apply Klingenberg's Theorem to obtain an injectivity
radius bound.
STEP 3. In Section 10, we derive a differential Harnack estimate of Li-
Yau- Hamilton type for the normalized Ricci flow. This is a lower bound for
the time derivative of the curvature
a 2
at log R - IV' log RI.(Again, there is a modified formula for the case that R (0) changes sign.)Integrating this along a space-time path from (x1, t1) to (x2, t2) such that 0 <
t 1 < t 2 yields a classical Harnack inequality, which is an a priori comparison
of the sort
STEP 4. In Section 11, we combine the differential Harnack estimate
with the bounds on IRI and the diameter derived in Step 2 to obtain a
positive lower bound for R. As outlined above, this allows us to prove the
desired convergence result for an initial metric of strictly positive curvature.
REMARK 5.36. In terms of getting a positive lower bound for R , the
BBS estimates are useful locally, whereas the differential Harnack estimate
is useful globally. Indeed, ifR ( X, t) = K, ~ Rmax ( t) > 0,
then the BBS estimates show that R (-, t) > r;,/2 in a g (t)-metric ball of
radius c/ .JK,, where c > 0 is a universal constant. On the other hand, if
the diameter is bounded, then the Harnack estimate yields a uniform lower
bound for R.7.2. The case that R (., 0) changes sign. To obtain convergence for
an arbitrary initial metric having r > 0, we prove that the scalar curvature
of any such solution eventually becomes everywhere positive. Once this
happens, the argument outlined above goes thiough.
In order to obtain suitable bounds on the curvature, we need the mod-
ified entropy formula and the modified Harnack estimate that apply when
the curvature is of mixed sign. We also need an ad hoc injectivity radius
estimate, since Klingenberg's Theorem may not apply if the curvature is
negative somewhere. We obtain an adequate estimate for the injectivity
radius in Section 12, and bound the curvature in Section 13.In Sections 14 and 15, we illustrate alternative methods for bounding
the injectivity radius and curvature of solutions whose curvature is initially
of mixed sign.