xxii CONTENTS OF PART II OF VOLUME TWO

the modified scalar curvature) is nonpositive:

r ( i ? + 2 A / - | V / |

2

) + / - 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.