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find canonical metrics on manifolds. The best known example of this is
the classical uniformization theorem for Riemann surfaces. Extremal
metrics were introduced by Calabi as an attempt at finding a
higher-dimensional generalization of this result, in the setting of
Kähler geometry.
This book gives an introduction to the study of
extremal Kähler metrics and in particular to the conjectural picture
relating the existence of extremal metrics on projective manifolds to
the stability of the underlying manifold in the sense of algebraic
geometry. The book addresses some of the basic ideas on both the
analytic and the algebraic sides of this picture. An overview is given
of much of the necessary background material, such as basic Kähler
geometry, moment maps, and geometric invariant theory. Beyond the basic
definitions and properties of extremal metrics, several highlights of
the theory are discussed at a level accessible to graduate students:
Yau's theorem on the existence of Kähler-Einstein metrics, the Bergman
kernel expansion due to Tian, Donaldson's lower bound for the Calabi
energy, and Arezzo-Pacard's existence theorem for constant scalar
curvature Kähler metrics on blow-ups.
