xviii CONTENTS OF PART I OF VOLUME TWO of view. These functionals will be considered in more detail in Chapters 5 and 6. Ricci solitons were actually introduced first in the physics literature, where nontrivial ones were called quasi-Einstein metrics. So it is perhaps not surprising that some aspect of duality theory is related to Ricci solitons. We discuss gradient Ricci solitons in the form of warped products with tori and the Buscher duality transformation of these special solitons. We conclude the chapter with a summary of results and open problems on Ricci solitons. A fundamental aspect of Ricci solitons is that they occur as singularity models, i.e., limits of dilations of a singular solution. In particular, for metrics with nonnegative curvature operators and whose scalar curvature attains its maximum in space and time, the limits of Type II singularities are steady Ricci solitons. One proof of this relies on the matrix differential Harnack inequality and the strong maximum principle for tensors, whose proofs are given in Part II. In the chapter on differential Harnack inequalities in Part II we shall motivate the consideration of the Harnack quadratic by differentiating the expanding gradient Ricci soliton equation to obtain the matrix Harnack quantity, which vanishes in certain directions on expanding gradient solitons. Chapter 2. We discuss the Kahler-Ricci flow, which is simply the Ricci flow on Kahler manifolds. In the compact case, the Ricci flow preserves the Kahler structure of the metric. Because of the interaction of the complex structure with the evolving metric, a rich field has developed in the study of the Kahler-Ricci flow. We begin by giving a basic introduction to Kahler geometry. This in- troduction is not meant as a replacement of the standard texts on Kahler geometry, but rather an attempt to make the book more self-contained. We encourage the novice to read other texts (some of these are cited in the notes and commentary) either before or in conjunction with this chapter. We emphasize local coordinate calculations in holomorphic coordinates in the style of the book by Morrow and Kodaira [275]. To put the study of the Kahler-Ricci flow in a broader context, we give a brief summary of some results on the existence and uniqueness of Kahler- Einstein metrics. Many of the results in this field are deep and we encourage the interested reader to consult the original papers or other sources. Our study of the Kahler-Ricci flow begins with the fundamental result of H.-D. Cao on the long-time existence and convergence on closed manifolds. Long-time existence holds independently of whether the first Chern class is negative, zero, or positive. Convergence to a Kahler-Einstein metric holds in the cases where the Chern class is either negative or zero. There has been substantial progress on the Kahler-Ricci flow in both the compact and the complete noncompact cases. We briefly survey some results in this area.
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