xxii CONTENTS OF PART II OF VOLUME TWO
the modified scalar curvature) is nonpositive:
r ( i ? + 2 A / - | V / |
) + / - n 0 .
Appendix D. In this appendix we review some basic results for the
Ricci flow. In particular, we recall the results on the short time and long time
existence and uniqueness of the Ricci flow on closed and noncompact mani-
folds, convergence results on closed manifolds assuming some sort of positiv-
ity of curvature, the rotationally symmetric neckpinch, curvature pinching
estimates, strong maximum principle, derivative estimates, differential Har-
nack estimates, Perelman's energy and entropy monotonicity and no local
collapsing, compactness theorems, and the existence of singularity models.
Appendix E. In this appendix we review some basic geometric analysis
related to the Ricci flow with an emphasis on the heat equation. We recall
Duhamel's principle and its application to basic results for the heat kernel.
We discuss the Cheeger-Yau comparison theorem for the heat kernel, the
Li-Yau differential Harnack estimate, and Hamilton's gradient estimates.
Appendix F. The material in this appendix is in preparation for Chap-
ter 15 on Hamilton's matrix Harnack estimate. Given a Riemannian mani-
fold, we describe a formalism for considering tensors as vector-valued func-
tions on the (orthonormal) frame bundle. In the context of the Ricci flow,
where we have a 1-parameter family of metrics, we add to this a modified
time derivative which may be considered as a version of Uhlenbeck's trick.
We discuss tensor calculus in this setting, including commutator formulas
for the heat operator and covariant derivatives. These calculations are used
for the Harnack calculations in Chapter 15.