172 5. THE RICCI FLOW ON SURFACES
in [ 28 ]. A new proof without the use of the potential function was given
by Hamilton and Yau [68]. The lower bound for the injectivity radius on
52 was proved by Hamilton. (See Section 5 of [28] or Section 12 of [63].)
The isoperimetric estimate in Section 14 was given by Hamilton [62]; the
application of this to give a new proof that R is uniformly bounded uses
the Gromov-type compactness theorem established in [64] and reviewed in
Section 3 of Chapter 7.
Results on the Ricci flow on noncompact surfaces are established in
[128], [129], [72], [73], and [35]. Convergence theorems on 2-orbifolds are
given in [127] and [34]. A new proof of the convergence on 52 using the
Aleksandrov method was obtained by Bartz, Struwe, and Ye [ 10 ]. A related
flow on 52 was studied by Leviton and Rubinstein [90].
The methods established for the Ricci flow are related to methods for
other geometric evolution equations. Both the entropy and differential Har-
nack estimates were established by Hamilton for the curve shortening flow
[44]. The entropy estimate was extended to higher dimensional hypersur-
face flows in [ 31 ] and [3]. References for differential Harnack inequalities of
LYH type in higher dimensions and for other geometric evolution equations
will be given in a chapter of the planned successor to this volume.