Value Distribution Theory of the Gauss Map of Minimal Surfaces in Rm

Hirotaka Fujimoto ist Professor am Institut für Mathematik der Kanazawa Universität in Japan.

• 1 The Gauss map of minimal surfaces in R3.- §1.1 Minimal surfaces in Rm.- §1.2 The Gauss map of minimal surfaces in Bm.- §1.3 Enneper-Weierstrass representations of minimal surfaces in R3.- §1.4 Sum to product estimates for meromorphic functions.- §1.5 The big Picard theorem.- §1.6 An estimate for the Gaussian curvature of minimal surfaces.- 2 The derived curves of a holomorphic curve.- §2.1 Holomorphic curves and their derived curves.- §2.2 Frenet frames.- §2.3 Contact functions.- §2.4 Nochka weights for hyperplanes in subgeneral position.- §2.5 Sum to product estimates for holomorphic curves.- §2.6 Contracted curves.- 3 The classical defect relations for holomorphic curves.- §3.1 The first main theorem for holomorphic curves.- §3.2 The second main theorem for holomorphic curves.- §3.3 Defect relations for holomorphic curves.- §3.4 Borel’s theorem and its applications.- §3.5 Some properties of Wronskians.- §3.6 The second main theorem for derived curves.- 4 Modified defect relation for holomorphic curves.- §4.1 Some properties of currents on a Riemann surface.- §4.2 Metrics with negative curvature.- §4.3 Modified defect relation for holomorphic curves.- §4.4 The proof of the modified defect relation.- 5 The Gauss map of complete minimal surfaces in Rm.- §5.1 Complete minimal surfaces of finite total curvature.- §5.2 The Gauss maps of minimal surfaces of finite curvature.- §5.3 Modified defect relations for the Gauss map of minimal surfaces.- §5.4 The Gauss map of complete minimal surfaces in R3 and R4.- §5.5 Examples.