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This book presents in a systematic and almost self-contained way the striking analogy between classical function theory, in particular the value distribution theory of holomorphic curves in projective space, on the one hand, and important and beautiful properties of the Gauss map of minimal surfaces on the other hand. Both theories are developed in the text, including many results of recent research. The relations and analogies between them become completely clear. The book is written for interested graduate students and mathematicians, who want to become more familiar with this modern development in the two classical areas of mathematics, but also for those, who intend to do further research on minimal surfaces.
Inhaltsverzeichnis
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.
Über den Autor / die Autorin
Hirotaka Fujimoto ist Professor am Institut für Mathematik der Kanazawa Universität in Japan.
