The theory of minimal surfaces is naturally linked to complex analysis. This book discusses the interaction between the theory of minimal surfaces and Nevanlinna theory. In particular, it describes the study of the value distribution properties of the Gauss map of minimal surfaces through Nevanlinna theory, a project initiated by the prominent differential geometers Shiing-Shen Chern and Robert Osserman.
Chapters 1 and 2 introduce mathematical foundations necessary to investigate minimal surfaces, including the basics of differential geometry and complex analysis. Chapter 1 includes the theory of curves in \(\mathbb{R}^2\) and \(\mathbb{R}^3\), the theory of surfaces in \(\mathbb{R}^3\), the Gauss-Bonnet theorem and the theory of abstract surfaces.
Chapter 2 includes the theory of holomorphic and harmonic functions, the Ahlfors-Schwarz lemma and the negative curvature method, and the theory of Riemann surfaces. Chapter 2 also includes the theory of algebraic curves, serving as an introduction to the theory of holomorphic curves (Nevanlinna theory) and playing a crucial rule in the study of Gauss maps of complete minimal surfaces with finite total curvature.
Chapter 3 focuses on the heart of the book, which is the value distribution of the Gauss maps of minimal surfaces immersed in \(\mathbb{R}^3\). Topics are minimal surfaces, the Weierstrass-Enneper representation, Gaussian maps of complete minimal surfaces in \(\mathbb{R}^3\), and minimal surfaces with finite total curvatures.
Chapter 4 gives a detailed presentation of the Nevanlinna theory: Nevanlinna theory of meromorphic functions, Cartan’s second main theorem for holomorphic curves, and holomorphic curves through Ahlfors’ method of negative curvature.
Chapter 5 studies the Gauss maps of minimal surfaces immersed in \(\mathbb{R}^m\). This chapter includes the theory of minimal surfaces in \(\mathbb{R}^m\), the Gauss curvature estimate, and the Gauss maps of minimal surfaces in \(\mathbb{R}^m\) with finite total curvature.
Chapter 6 establishes the nonintegrated defect relation for the meromorphic maps of complete Kähler manifolds, whose universal cover is the ball \(B(R_0)\) (\(0<R_0\leq\infty\)), into \(\mathbb{P}^N(\mathbb{C})\) following Xavier’s method developed in his proof of the “seven-point theorem”.
The book is intended to be self-contained with few prerequisites, making it suitable for undergraduate students as well as graduate students and researchers.