# Contributions to Analysis

### A Collection of Papers Dedicated to Lipman Bers

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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

Preface

On the Decomposition and Deformation of Kleinian Groups

Introduction

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

References

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

References

Conditions for Quasiconformal Deformations in Several Variables

1. Introduction

2. Uniqueness

3. The Adjoint Operator

4. Statement of the Theorem

5. Necessity

6. Sufficiency

References

On Locally Quasiconformal Mappings in Space (n ≥ 3)

References

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

Introduction

1. Boundary Correspondence Theorem

2. Application to Cavity Flows

References

The Extension Problem for Quasiconformal Mappings

References

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)

References

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)

Lecture

References

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

References

On Holomorphic Mappings between Teichmüller Spaces

Introduction

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

References

On the Poincaré Relation

Introduction

References

Elliptic Functions and Modular Forms

Introduction

Reference

On the Differentiability of Solutions of Accretive Linear Differential Equations

References

Survey of Some Recent Progress in Transonic Aerodynamics

1. Introduction

2. Optimal Transonic Airfoils

3. Three-Dimensional Problems

References

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

References

Two Results in the Global Theory of Holomorphic Mappings

Introduction

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

References

On Fundamental Domains and the Teichmüller Modular Group

1. Introduction

2. The Torus with a Hole

3. Geodesies on S

4. The Twist Parameter

References

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

Introduction

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

References

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

Introduction

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

References

Group Isomorphisms Induced by Quasiconformal Mappings

References

Partial Differential Equations Invariant under Conformai or Projective Transformations

Introduction

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

References

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

References

Homomorphisms Associated with Multiplicative Functions

Introduction

1. Complex Tori and Multiplicative Functions

2. α(Φ,F)

References

Uniformizations of Riemann Surfaces

Introduction

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

References

Some Restrictions on the Smooth Immersion of Complete Surfaces in E3

Introduction

1. Definitions

2. Preliminary Results

3. Some Theorems

4. Subsidiary Results

References

Prym Varieties I

Introduction

Notations

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

References

Asymptotic Decay for Ultrahyperbolic Operators

1. Introduction

2. The Ultrahyperbolic Operator

3. Generalized Ultrahyperbolic Operators

References

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)

References

Extremal Quasiconformal Mappings with Given Boundary Values

Introduction

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

References

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

References

A Constructive Proof of the Riemann-Roch Theorem for Curves

References

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

References

## Product details

- No. of pages: 460
- Language: English
- Copyright: © Academic Press 1974
- Published: January 1, 1974
- Imprint: Academic Press
- eBook ISBN: 9781483261164