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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.

Analytic functions. --- Functions, Meromorphic. --- Value distribution theory. --- Meromorphic functions --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Functions of complex variables --- Series, Taylor's --- Distribution of values theory --- Functions, Entire --- Functions, Meromorphic --- Addition. --- Algebraic curve. --- Algebraic number. --- Atlas (topology). --- Binomial coefficient. --- Cauchy–Riemann equations. --- Compact Riemann surface. --- Compact space. --- Complex manifold. --- Complex projective space. --- Computation. --- Continuous function (set theory). --- Covariant derivative. --- Critical value. --- Curvature form. --- Diagram (category theory). --- Differential form. --- Differential geometry of surfaces. --- Differential geometry. --- Dimension. --- Divisor. --- Essential singularity. --- Euler characteristic. --- Existential quantification. --- Fiber bundle. --- Gaussian curvature. --- Geodesic curvature. --- Geometry. --- Grassmannian. --- Harmonic function. --- Hermann Weyl. --- Hermitian manifold. --- Holomorphic function. --- Homology (mathematics). --- Hyperbolic manifold. --- Hyperplane. --- Hypersurface. --- Improper integral. --- Intersection number (graph theory). --- Isometry. --- Line integral. --- Manifold. --- Meromorphic function. --- Minimal surface. --- Nevanlinna theory. --- One-form. --- Open problem. --- Open set. --- Orthogonal complement. --- Parameter. --- Picard theorem. --- Product metric. --- Q.E.D. --- Remainder. --- Riemann sphere. --- Riemann surface. --- Smoothness. --- Special case. --- Submanifold. --- Subset. --- Tangent space. --- Tangent. --- Theorem. --- Three-dimensional space (mathematics). --- Unit circle. --- Unit vector. --- Vector field. --- Volume element. --- Volume form. --- Fonctions de plusieurs variables complexes