The Equidistribution Theory of Holomorphic Curves. (AM-64), Volume 64 (Annals of Mathematics Studies)

This work is a fresh presentation of the Ahlfors-Weyl theory of holomorphic curves that takes into account some recent developments in Nevanlinna theory and several complex variables. The treatment is differential geometric throughout, and assumes no previous acquaintance with the classical theory of Nevanlinna. The main emphasis is on holomorphic curves defined over Riemann surfaces, which admit a harmonic exhaustion, and the main theorems of the subject are proved for such surfaces. The author discusses several directions for further research. CONTENTS Preface Introduction Chapter I. Generalities on projective spaces and Grassmannians §1 The Fubini-Study metric §2 Basic facts about Grassmannians §3 Elementary inequalities Chapter II. Nevanlinna theory of meromorphic functions §1 The proximity function §2 Holomorphic mappings into the Riemann sphere §3 The nonintegrated main theorems §4 Harmonic exhaustion and the integrated main theorems §5 Two useful facts §6 The defect relation Chapter III. Elementary properties of holomorphic curves §1 Holomorphic curves induced by a system of holomorphic functions §2 Nondegeneracy §3 Associated curves §4 Projection curves §5 Contracted curves of the first kind §6 Contracted curve of the second kind Chapter IV. The two main theorems for holomorphic curves §1 The nonintegrated first main theorem §2 The integrated first main theorem §3 The FMT of rank k §4 The generalized FMT and two inequalities §5 The nonintegrated second main theorem and PIücker's formulas §6 The integrated second main theorem §7 The SMT of rank k Chapter V. The defect relations §1 A basic theorem on integration over PnC §2 The fundamental inequality §3 The fundamental inequality of arbitrary rank §4 Proof of the defect relations §5 Applications References Index of principal definitions