The Geometric Theory of Complex Variables / by Peter V. Dovbush, Steven G. Krantz.

Format:

Book

Author/Creator:

Dovbush, Peter V.

Contributor:

Krantz, Steven G.

Language:

English

Subjects (All):

Global analysis (Mathematics).

Manifolds (Mathematics).

Functions of complex variables.

Functional analysis.

Global Analysis and Analysis on Manifolds.

Several Complex Variables and Analytic Spaces.

Functional Analysis.

Local Subjects:

Global Analysis and Analysis on Manifolds.

Several Complex Variables and Analytic Spaces.

Functional Analysis.

Physical Description:

1 online resource (1074 pages)

Edition:

1st ed. 2025.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2025.

Summary:

This book provides the reader with a broad introduction to the geometric methodology in complex analysis. It covers both single and several complex variables, creating a dialogue between the two viewpoints. Regarded as one of the 'grand old ladies' of modern mathematics, complex analysis traces its roots back 500 years. The subject began to flourish with Carl Friedrich Gauss's thesis around 1800. The geometric aspects of the theory can be traced back to the Riemann mapping theorem around 1850, with a significant milestone achieved in 1938 with Lars Ahlfors's geometrization of complex analysis. These ideas inspired many other mathematicians to adopt this perspective, leading to the proliferation of geometric theory of complex variables in various directions, including Riemann surfaces, Teichmüller theory, complex manifolds, extremal problems, and many others. This book explores all these areas, with classical geometric function theory as its main focus. Its accessible and gentle approach makes it suitable for advanced undergraduate and graduate students seeking to understand the connections among topics usually scattered across numerous textbooks, as well as experienced mathematicians with an interest in this rich field.

Contents:

- Introduction

The Riemann Mapping Theorem

The Ahlfors Map

A Riemann Mapping Theorem for Two-Connected Domains in the Plane

Riemann Multiply Connected Domains

Quasiconformal Mappings

Manifolds

Riemann Surfaces

The Uniformization Theorem

Automorphism Groups

Ridigity of Holomorphic Mappings and a New Schwarz Lemma at the Boundary

The Schwarz Lemma and Its Generalizations

Invariant Distances on Complex Manifolds

Hyperbolic Manifolds

The Fatou Theory and Related Matters

The Theorem of Bun Wong and Rosay

Smoothness to the Boundary of Biholomorphic Mappings

Solution ∂ problem

Harmonic measure

Quadrature

Teichmüller Theory

Bibliography

Index.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9783031772047

3031772040

OCLC:

1499722222