微分几何中的一个基本问题是在流形上寻找正则度量。最著名的例子是Riemann面的经典单值化定理。Calabi引入极值度量是为了在K?hler几何的框架中找到这一结果的高维推广。 本书介绍了对极值K?hler度量的研究，特别是关于射影流形上极值度量的存在与代数几何意义下的基本流形的稳定性猜想。本书阐述了猜想在分析和代数两方面的一些基本思想；概述了许多必要的背景材料，如基本K?hler几何、矩映射和几何不变理论。除了极值度量的基本定义和性质之外，本书也对该理论的几个亮点在研究生可以理解的水平上进行了讨论：关于K?hler-Einstein度量存在性的丘成桐定理、田刚的Bergman核展开、Donaldson的Calabi能量下界以及爆破的常标量曲率K?hler度量的Arezzo-Pacard存在定理。

前辅文
Chapter 1. K¨ahler Geometry
  1.1. Complex manifolds
  1.2. Almost complex structures
  1.3. Hermitian and K¨ahler metrics
  1.4. Covariant derivatives and curvature
  1.5. Vector bundles
  1.6. Connections and curvature of line bundles
  1.7. Line bundles and projective embeddings
Chapter 2. Analytic Preliminaries
  2.1. Harmonic functions on Rn
  2.2. Elliptic differential operators
  2.3. Schauder estimates
  2.4. The Laplace operator on K¨ahler manifolds
Chapter 3. K¨ahler-Einstein Metrics
  3.1. The strategy
  3.2. The C0- and C2-estimates
  3.3. The C3- and higher-order estimates
  3.4. The case c1(M) = 0
  3.5. The case c1(M) >0
  3.6. Futher reading
Chapter 4. Extremal Metrics
  4.1. The Calabi functional
  4.2. Holomorphic vector fields and the Futaki invariant
  4.3. The Mabuchi functional and geodesics
  4.4. Extremal metrics on a ruled surface
  4.5. Toric manifolds
Chapter 5. Moment Maps and Geometric Invariant Theory
  5.1. Moment maps
  5.2. Geometric invariant theory (GIT)
  5.3. The Hilbert-Mumford criterion
  5.4. The Kempf-Ness theorem
  5.5. Relative stability
Chapter 6. K-stability
  6.1. The scalar curvature as a moment map
  6.2. The Hilbert polynomial and flat limits
  6.3. Test-configurations and K-stability
  6.4. Automorphisms and relative K-stability
  6.5. Relative K-stability of a ruled surface
  6.6. Filtrations
  6.7. Toric varieties
Chapter 7. The Bergman Kernel
  7.1. The Bergman kernel
  7.2. Proof of the asymptotic expansion
  7.3. The equivariant Bergman kernel
  7.4. The algebraic and geometric Futaki invariants
  7.5. Lower bounds on the Calabi functional
  7.6. The partial C0-estimate
Chapter 8. CscK Metrics on Blow-ups
  8.1. The basic strategy
  8.2. Analysis in weighted spaces
  8.3. Solving the non-linear equation when n > 2
  8.4. The case when n = 2
  8.5. The case when M admits holomorphic vector fields
  8.6. K-stability of cscK manifolds
Bibliography
Index