332 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS
original papers [135] and [134] for both the parabolic and elliptic versions
of Moser iteration.
For Theorem 25.2 see Lemma 3.1 in Chau, Tam, and Yu [26], which is
based on §5 of Zhang [195].
For Proposition 25.4, see Theorem 3.1 of Saloff-Coste [165]. See Gallot
[65] for earlier work and see Aubin [8], Hebey [96], and Hebey and Vaugon
[98] for sharp forms.
§2. Originally, for linear second-order parabolic operators in divergence
form on Euclidean space, Harnack estimates have been proved by Moser
iteration (see Moser [135]). On Riemannian manifolds, Li and Yau [121]
proved a Harnack estimate by integrating a gradient estimate along space-
time paths. In Saloff-Coste [165], partly based on the observation that
qualitatively, Sobolev constants on Riemannian manifolds depend only on
their rough isometry class, i.e., if g :::; Cg, then the Sobolev constants of
g and g differ at most by a factor depending only on C and n, a Harnack
estimate on Riemannian manifolds was proved by Moser iteration.
In Chapters 15 and 16 of Part II we discussed Hamilton's matrix Harnack
estimate for the Ricci flow and Perelman's differential Harnack estimate for
the adjoint heat equation coupled to the Ricci flow, respectively.
For Theorem 25.8 see Li and Yau [121] and Lemma 4.1 of Chau, Tam,
and Yu [26].