by Ngaiming Mok (Columbia Univ.)

This monograph studies the problem of characterizing canonical metrics on Hermitian locally symmetric manifolds X of non-compact/compact types in terms of curvature conditions. The proofs of these metric rigidity theorems are applied to the study of holomorphic mappings between manifolds X of the same type. Moreover, a dual version of the generalized Frankel Conjecture on characterizing compact Kähler manifolds are also formulated.

• Background and First Results: Historical Background and Summary of Results
• Fundamentals of Hermitian and Kähler Geometries
• Riemannian and Hermitian Symmetric Manifolds
• Bounded Symmetric Domains — the Classical Cases
• Bounded Symmetric Domains — General Theory
• The Hermitian Metric Rigidity Theorem for Compact Quotients
• The Kähler Metric Rigidity Theorem in the Semipositive Case
• Further Development: The Hermitian Metric Rigidity Theorem for Quotients of Finite Volume
• The Immersion Problem for Complex Hyperbolic Space Forms
• The Hermitian Metric Rigidity Theorem on Locally Homogeneous Holomorphic Vector Bundles
• A Rigidity Theorem for Holomorphic Mappings between Irreducible Hermitian Symmetric Manifolds of Compact Type
• Appendix: Semisimple Lie Algebras and Their Representations
• Some Theorems in Riemannian Geometry
• Characteristic Projective Subvarieties Associated to Hermitian Symmetric Manifolds
• A Dual Generalized Frankel Conjecture for Compact Kähler Manifolds of Seminegative Bisectional Curvature