CONTENTS OF PART I OF VOLUME TWO xix There are a number of interesting Kahler-Ricci solitons. We present the construction of some of these solitons, including the Koiso soliton. All of the known examples have some sort of symmetry. It is hoped that the study of Kahler-Ricci solitons may shed some light on the problem of formulating weak solutions to the Kahler-Ricci flow as a way of canonically flowing past a singularity. Finally we discuss nonnegativity of curvature and the Kahler-Ricci flow. A particularly natural condition is nonnegative bisectional curvature. This condition is preserved under the Kahler-Ricci flow and H.-D. Cao has proved a differential matrix Harnack estimate under this curvature condition (as- suming bounded curvature). The trace form of the matrix estimate may be generalized to a differential Harnack estimate which ties more closely to the heat equation. We present a family of such inequalities and discuss some applications. Chapter 3. The study of the limiting behavior of solutions begins with Hamilton's Cheeger-Gromov-type compactness theorem for the Ricci flow, which is the subject of this chapter. One considers sequences of pointed solutions to the Ricci flow and one attempts to extract a limit of a subse- quence. Such sequences arise when studying singular solutions by taking sequences of points and times approaching the singularity time and dilating the solutions by the curvatures at these basepoints. In order to extract a limit, we assume that the injectivity radii at the basepoints and the cur- vatures everywhere are bounded. By Shi's local derivative bounds, we get pointwise bounds on all the derivatives of the curvatures. This enables us to prove convergence in C°° on compact subsets for a subsequence. We prove the compactness theorem for solutions (time-dependent) from the compactness theorem for metrics (time-independent), which will be proved in the next chapter. We also consider a local version of the com- pactness theorem and discuss the application of the compactness theorem to the existence of singularity models for solutions of the Ricci flow assum- ing an injectivity radius estimate (such an estimate holds for finite time solutions on closed manifolds by Perelman's no local collapsing theorem). The outline of the proof of the compactness theorem is as follows. One first proves a compactness theorem for pointed Riemannian manifolds with bounded injectivity radii, curvatures and derivatives of curvature. To prove the compactness theorem for pointed solutions {M,gk (t), Ok}ke^ to the Ricci flow from this, one observes the following. By Shi's estimate and the compactness theorem for pointed Riemannian manifolds there exists a sub- sequence gk (to) which converges for a fixed time to- The bounds on the curvatures and their derivatives also imply that the metrics g^ (t) are uni- formly equivalent on compact time intervals and the covariant derivatives of the metrics gk (t) with respect to a fixed metric g are bounded. The com- pactness theorem for solutions then follows from the Arzela-Ascoli theorem. We also briefly discuss the Cheeger-Gromov-type compactness theorems for both Kahler metrics and solutions of the Kahler-Ricci flow. The only
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