Contributions to Analysis - 1st Edition - ISBN: 9780120448500, 9781483261164

Contributions to Analysis

1st Edition

A Collection of Papers Dedicated to Lipman Bers

Editors: Lars V. Ahlfors Irwin Kra Bernard Maskit
eBook ISBN: 9781483261164
Imprint: Academic Press
Published Date: 1st January 1974
Page Count: 460
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Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers is a compendium of papers provided by Bers, friends, students, colleagues, and professors. These papers deal with Teichmuller spaces, Kleinian groups, theta functions, algebraic geometry. Other papers discuss quasiconformal mappings, function theory, differential equations, and differential topology. One paper discusses the results of the rigidity theorem of Mostow and its generalization by Marden in relation to geometric properties of Kleinian groups of the first kind. These results, obtained by planar methods, are presented in terms of the hyperbolic 3-space language, which is a natural pedestal in approaching the action of the Kleinian groups. Another paper reviews Riemann's vanishing theorem which solves the Jacobi inversion problem, by relating the vanishing properties of the theta function (particularly at half periods) to properties of certain linear series on the Riemann surface. One paper examines the problem of obtaining relations among the periods of the differentials of first kind on a compact Riemann surface. An application of a computer program involves supersonic transport. The program is based on the hodograph transformation and a method of complex characteristics to calculate profiles that are shock-less at a specified angle of attack, or at a specified subsonic free-stream Mach number. The collection can prove useful for engineers, statisticians, students, and professors in advance mathematics or courses related to aeronautics.

Table of Contents

List of Contributors


On the Decomposition and Deformation of Kleinian Groups


1. Some Kleinian Groups

2. The Maskit Combination Theorems

3. The Decomposition of Finitely Generated Kleinian Groups of the Second Kind

4. Deformations of Finitely Generated Kleinian Groups


Some Loci of Teichmüller Space for Genus Five Defined by Vanishing Theta Nulls

1. Introduction

2. Riemann's Vanishing Theorem

3. g31s on Surfaces of Genus Five

4. Surfaces of Genus Five Admitting Two Half-Canonical g41s

5. Proofs of the Theorems

6. Necessary Conditions for Elliptic-Hyperellipticity

7. Some Remarks on the "p — 2 Conjecture" for Higher Genus


Conditions for Quasiconformal Deformations in Several Variables

1. Introduction

2. Uniqueness

3. The Adjoint Operator

4. Statement of the Theorem

5. Necessity

6. Sufficiency


On Locally Quasiconformal Mappings in Space (n ≥ 3)


A Theorem on the Boundary Correspondence under Conformal Mapping with Application to Free Boundary Problems of Fluid Dynamics


1. Boundary Correspondence Theorem

2. Application to Cavity Flows


The Extension Problem for Quasiconformal Mappings


Hyperbolic Spaces

1. Introduction and Preliminaries

2. The Hyperbolic Space Hn(F)

3. Conjugacy Classes in U(1,n;F)

4. Subgroups of U(1,n;F)


On Σ-Monogenic Functions, and the Mean Value Theorem of the Differential Calculus

1. Reminiscence, and Σ-Monogenic Functions

2. Remarks on the Elementary Mean Value Theorem of the Differential Calculus

Abstract (of Lecture)



On Quasiconformal Extensions of the Beurling-Ahlfors Type

1. Introduction

2. Statement of Theorems

3. Proof of Theorem 1

4. Proof of Theorem 2

5. Proof of Theorem 3


On Holomorphic Mappings between Teichmüller Spaces


1. Teichmüller Spaces

2. The Teichmüller Metric

3. The Automorphism Group of T(g,n)

4. The Teichmüller Curve and Its Sections

5. Most Fiber Spaces Are Not Teichmüller Spaces


On the Poincaré Relation



Elliptic Functions and Modular Forms



On the Differentiability of Solutions of Accretive Linear Differential Equations


Survey of Some Recent Progress in Transonic Aerodynamics

1. Introduction

2. Optimal Transonic Airfoils

3. Three-Dimensional Problems


The Hausdorff Measure of Sets Which Link in Euclidean Space

1. Introduction

2. Preliminary Results

3. Main Construction

4. Main Results

5. An Inequality for Quasiconformal Mappings


Two Results in the Global Theory of Holomorphic Mappings


1. Statement of Theorem I

2. The First Main Theorem; A Result of Chern-Wu

3. Two Estimates from the Theory of Holomorphic Curves

4. Proof of Theorem I

5. Some Comments and Examples

6. Statement and Proof of Theorem II

7. Some Comments and Questions Regarding Holomorphic Curves


On Fundamental Domains and the Teichmüller Modular Group

1. Introduction

2. The Torus with a Hole

3. Geodesies on S

4. The Twist Parameter


Differential Equations in a Projective Space and Linear Dependence over a Projective Variety


1. Canonical Characteristic Sets

2. Differentially Homogeneous and Differentially Multihomogeneous Differential Polynomials

3. Differentially Homogeneous and Differentially Multihomogeneous Differential Ideals

4. Algebraic Differential Equations in Projective and Multiprojective Spaces

5. Differential Specializations

6. Differentially Complete Differentially Closed Sets

7. Linear Dependence over Projective Varieties

8. Examples


On the Complexes on the Boundary Induced by Elliptic Complexes of Differential Operators


0. Preliminaries

1. Normal Derivative Operators

2. Operators Induced on the Boundary

3. Operators on the Boundary Induced by the Elliptic Complex of Differential Operators


Group Isomorphisms Induced by Quasiconformal Mappings


Partial Differential Equations Invariant under Conformai or Projective Transformations


Part I

1. Derivation of Conditions (4) and (5) for Partial Conformal Invariance of (3)

2. The Dirichlet Problem for (6′) in Domains with Smooth Compact Boundaries

3. A Priori Estimates for Nonnegative Solutions of (2.1)

4. Solutions of (6′) Which Are Infinite on the Boundary

5. Solutions of (6′) for Arbitrary Domains in Rn

Part II

6. Invariance of (9) and (10) under Projective Transformations

7. The Boundary Value Problem (9)

8. A Priori Estimates for Second Derivatives

9. The Metric (10) Is Complete

10. The Mean Curvature of the Surface Is Bounded


Schottky Groups and Circles

1. Introduction

2. Schottky Space

3. Classical Schottky Space

4. Convergence of Classical Schottky Groups

5. Proof of the Proposition

6. Proof of Theorem 3.1

7. Fuchsian Groups of the Second Kind


Homomorphisms Associated with Multiplicative Functions


1. Complex Tori and Multiplicative Functions

2. α(Φ,F)


Uniformizations of Riemann Surfaces


1. What Is a Uniformization?

2. Isomorphisms

3. Factor Subgroups

4. Precisely Invariant Sets

5. Picture of a Uniformization

6. Standard Uniformizations

7. Torsion-Free Uniformizations

8. The General Uniformization Theorem


Some Restrictions on the Smooth Immersion of Complete Surfaces in E3


1. Definitions

2. Preliminary Results

3. Some Theorems

4. Subsidiary Results


Prym Varieties I



Part I

1. Double Coverings of Curves

2. A Configuration of Abelian Varieties

3. Definition of the Prym Variety

Part II

4. Relations between Theta Divisors

5. The Splitting of φ-1(θy,y) for Jacobians

Part III

6. Geometric Description of Sing Ξ, Unramified Case

7. Dim Sing Ξ

Appendix : A Theorem of Martens


Asymptotic Decay for Ultrahyperbolic Operators

1. Introduction

2. The Ultrahyperbolic Operator

3. Generalized Ultrahyperbolic Operators


Instability of Thin-Walled Spherical Structures under External Pressure

1. Introduction

2. Basic Equations

3. Method of Solution of the Equations

4. A Numerical Example

5. Legendre Functions and the Coefficients (l,m,n)


Extremal Quasiconformal Mappings with Given Boundary Values


1. Quasiconformal Mappings of the Unit Disc with the Same Boundary Values

2. The Sufficiency of Hamilton's Condition

3. Proof of Hamilton's Theorem (Necessity for Extremality)

4. Some Consequences of Hamilton's Condition


Invariant Metrics on Teichmüller Space

1. Differential Metrics on Teichmüller Space

2. Hermitian Metrics on Tg

3. Metrics Associated with the Embedding of Tg in the Siegel Upper Half-Plane


A Constructive Proof of the Riemann-Roch Theorem for Curves


Function Theory on Differentiable Submanifolds

1. Introduction

2. Holomorphic Approximation and Extension: Examples

3. Holomorphic Approximation

4. Envelopes of Holomorphy and Holomorphic Extension

5. CR Function Theory



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© Academic Press 1974
Academic Press
eBook ISBN:

About the Editor

Lars V. Ahlfors

Irwin Kra

Bernard Maskit

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