Chapter 2. Kahler-Ricci Flow and Kahler-Ricci Solitons
Symmetry, as wide or narrow as you may define its meaning, is one idea by
which man through the ages has tried to comprehend and create order, beauty,
and perfection. -Hermann WeylThe Kahler-Ricci fl.ow is simply an abbreviation for the Ricci fl.ow on
Kahler manifolds. In this chapter we first review sorne basic definitions and
properties for Kahler manifolds and survey sorne of the fundamental results
on the existence of Kahler-Einstein metrics. Then we discuss elementary
properties of the Kahler-Ricci fl.ow and state sorne of the fundamental long-
time existence and convergence results. We also give a survey of Kahler-
Ricci solitons.
Sarne other highlights of this chapter are an exposition of tensor cal-
culations in holornorphic coordinates, the proof of the long-tirne existenceand convergence of the Kahler-Ricci fl.ow on Kahler manifolds with c 1 < 0,
construction of the Koiso solitons and other U(n)-invariant solitons, proofs
of differential Harnack estimates under the assumption of nonnegative holo-
rnorphic bisectional curvature, and a survey of uniforrnization-type results
for complete noncornpact Kahler manifolds with positive bisectional curva-
ture.
We assume the reader either has sorne knowledge of Kahler geometry or
will read other references on Kahler geometry along with this chapter, so we
rnake no attempt to be completely self-contained.^1 The latter chapters do
not depend on this chapter and the reader interested only in Riemannian
Ricci fl.ow rnay skip this chapter.
1. Introduction to Kahler manifolds
In this section we introduce the basic concepts of complex and Kahler
manifolds including the point of view of having an almost complex structure
on an even-dimensional Riernannian manifold satisfying natural properties.
Let M be a real 2n-dirnensional differentiable manifold. A system of
holomorphic coordinates on M is a collection {zi : ui ---t Zi (Ui) c en}'
where {Ui} is a cover of M and Zi are horneornorphisrns such that the rnaps
zi o zj^1 : Zj (Ui n Uj) ___, zi (Ui n Uj)
(^1) See the notes and commentary at the end of this chapter for some references on
complex manifolds and Kahler geometry.
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