Schwarz's lemma from a differential geometric viewpoint

• Main Authors: Kim, Kang-Tae, Lee, Hanjin
• Format: eBook Book
• Language: English
• Published: Singapore World Scientific Publishing Co. Pte. Ltd 2010; Bangalore, India IISc Press; World Scientific; World Scientific Publishing Company; Co-Published with Indian Institute of Science (IISc), Bangalore, India; World Scientific Publishing
• Edition: 1
• Series: IISc lecture notes series
• Subjects: Complex Analysis; Geometry, Riemannian; Holomorphic functions; Holomorphic mappings; Mathematics; Pure Mathematics; Schwarz function; SCIENCE; Manifolds; Hermitian; Schwarz's Lemma; Maximum Principle; Curvature; Holomorphic
• ISBN: 9789814324786; 9814324787; 9814324795; 9789814324793
• DOI: 10.1142/7944

The subject matter in this volume is Schwarz's Lemma which has become a crucial theme in many branches of research in mathematics for more than a hundred years to date. This volume of lecture notes focuses on its differential geometric developments by several excellent authors including, but no...

Table of Contents:

• Schwarz's lemma from a differential geometric viewpoint -- Series Preface -- Preface -- Contents -- Chapter 1: Some Fundamentals -- Chapter 2: Classical Schwarz's Lemma and the Poincaré Metric -- Chapter 3: Ahlfors' Generalization -- Chapter 4: Fundamentals of Hermitian and Kählerian Geometry -- Chapter 5: Chern-Lu Formulae -- Chapter 6: Tamed Exhaustion and Almost Maximum Principle -- Chapter 7: General Schwarz's Lemma by Yau and Royden -- Chapter 8: More Recent Developments -- Bibliography -- Index.
• Intro -- Contents -- Series Preface -- Preface -- Chapter 1 Some Fundamentals -- 1.1 Mean-Value Property -- 1.2 Maximum Principle, I―Harmonic and Holomorphic Functions -- 1.3 Maximum Principle, II―For Subharmonic Functions -- Chapter 2 Classical Schwarz's Lemma and the Poincaré Metric -- 2.1 Classical Schwarz's Lemma -- 2.2 Pick's Generalization -- 2.3 The Poincaré Length and Distance -- Chapter 3 Ahlfors' Generalization -- 3.1 Generalized Schwarz's Lemma by Ahlfors -- 3.2 Application to Kobayashi Hyperbolicity -- Chapter 4 Fundamentals of Hermitian and Kählerian Geometry -- 4.1 Almost Complex Structure -- 4.2 Tangent Space and Bundle -- 4.3 Cotangent Space and Bundle -- 4.3.1 Hermitian metric -- 4.4 Connection and Curvature -- 4.4.1 Riemannian connection and curvature -- 4.4.2 Riemann curvature tensor and sectional curvature -- 4.4.3 Holomorphic sectional curvature -- 4.4.4 The case of Poincaré metric of the unit disc -- 4.5 Connection and Curvature in Moving Frames -- 4.5.1 Hermitian metric, frame and coframe -- 4.5.2 Hermitian connection -- 4.5.3 Curvature -- 4.5.4 The Hessian and the Laplacian -- Chapter 5 Chern-Lu Formulae -- 5.1 Pull-Back Metric against the Original -- 5.2 Connection, Curvature and Laplacian -- 5.3 Chern-Lu Formulae -- 5.4 General Schwarz's Lemma by Chern-Lu -- Chapter 6 Tamed Exhaustion and Almost Maximum Principle -- 6.1 Tamed Exhaustion -- 6.2 Almost Maximum Principle -- Chapter 7 General Schwarz's Lemma by Yau and Royden -- 7.1 Generalization by S.T. Yau -- 7.2 Schwarz's Lemma for Volume Element -- 7.3 Generalization by H.L. Royden -- Chapter 8 More Recent Developments -- 8.1 Osserman's Generalization -- 8.2 Schwarz's Lemma for Riemann Surfaces with K ≤ 0 -- 8.3 Final Remarks -- Bibliography -- Index