Series:Springer Proceedings in Mathematics & Statistics, 349Periodical:

Publisher:SpringerCity:Cham

Year:2021Edition:

Language:EnglishPages (bibliotech):288281

ISBN:3030685403, 9783030685409ID:2979177

Time added:2021-05-10 15:48:57Time modified:2021-05-10 19:41:22

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Library issue:

Size:7 MB (6888698 bytes)Extension:pdf

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Topic:MathematicsTags:

This book collects original peer-reviewed contributions to the conferences organised by the international research network “Minimal surfaces: Integrable Systems and Visualization” financed by the Leverhulme Trust. The conferences took place in Cork, Granada, Munich and Leicester between 2016 and 2019. Within the theme of the network, the presented articles cover a broad range of topics and explore exciting links between problems related to the mean curvature of surfaces in homogeneous 3-manifolds, like minimal surfaces, CMC surfaces and mean curvature flows, integrable systems and visualisation. Combining research and overview articles by prominent international researchers, the book offers a valuable resource for both researchers and students who are interested in this research area.

Table of contents :

Introduction

Proceedings of the Workshop Series of Minimal Surfaces: Integrable Systems and Visualisation

Contents

Translating Solutions to Mean Curvature Flow

1 Introduction

2 Translators

3 Ancient Solutions with Discrete Symmetry Groups

References

Finding Conformal and Isometric Immersions of Surfaces

1 Introduction

2 Spin Bundles and Regular Homotopy Classes

3 Conformal and Isometric Immersions

References

Discrete Minimal Nets with Symmetries

1 Introduction

2 Preliminaries

2.1 Discrete Isothermic Nets

2.2 Discrete Gaussian and Mean Curvatures

2.3 Planar Reflection Principle for Discrete Isothermic Minimal and cmc Nets

3 Reflection Properties of Discrete Minimal Nets

3.1 Discrete Isothermic Minimal Nets

3.2 Discrete Asympotic Minimal Nets

3.3 Reflection Properties of Discrete Minimal Nets

4 Examples of Discrete Minimal Nets with Symmetry

References

A Survey on Minimal Isometric Immersions into mathbbR3, mathbbS2timesmathbbR and mathbbH2timesmathbbR

1 Introduction

2 The Local Problem in mathbbR3

2.1 The Associate Family

2.2 Calabi’s Rigidity Result

2.3 The Existence Problem

2.4 Generalizations

3 The Global Problem in mathbbR3

4 The Local Problem in mathbbS2timesmathbbR and mathbbH2timesmathbbR

5 Possible Extensions

References

Families of Minimal Surfaces in mathbbH2 timesmathbbR Foliated by Arcs and Their Jacobi Fields

1 Introduction

2 Preliminaries and Notation

3 Catenoids of Revolution in mathbbH2 timesmathbbR

4 Parabolic Catenoids

5 Tall Rectangles

6 Jacobi Fields on Parabolic Catenoids

6.1 Deformations to Catenoids

6.2 Deformations to Tall Rectangles

7 The Jacobi Operator on Parabolic Catenoids

References

Jenkins–Serrin Graphs in MtimesmathbbR

1 Introduction

2 Compactness Theorem for Stable Integral Varifolds

3 Translating Graphs

3.1 The Euclidean Case

4 Jenkins–Serrin Translating Solitons

References

Survey on the Asymptotic Dirichlet Problem for the Minimal Surface Equation

1 Preliminaries

1.1 Cartan–Hadamard Manifolds

1.2 Asymptotic Dirichlet Problem on Cartan–Hadamard Manifolds

2 Overview of the Results

2.1 Minimal Surfaces

2.2 Other Prescribed Mean Curvature Surfaces

3 Different Strategies to Solve the Asymptotic Dirichlet Problem

3.1 Methods Related to Barriers

3.2 Sobolev and Poincaré Inequalities with Moser Iteration

4 Non-solvability of the Asymptotic Dirichlet Problem

References

Generalized Whitham Flow and Its Applications

1 Introduction

2 The Deligne–Hitchin Moduli Space

2.1 Stability

2.2 Automorphisms of the Deligne–Hitchin Moduli Space

3 Higher Genus CMC Surfaces

3.1 Abelianization

3.2 Deformation of Harmonic Maps

4 Higher Solutions and Constrained Willmore Surfaces

References

Notes on Translating Solitons for Mean Curvature Flow

1 Introduction

2 Some Remarks About Singularities

3 Translators

3.1 Variational Approach

4 Examples of Translators

5 Graphical Translators

6 The Spruck–Xiao Convexity Theorem

7 Characterization of Translating Graphs in R3

References

Uniqueness Problem for Closed Non-smooth Hypersurfaces with Constant Anisotropic Mean Curvature and Self-similar Solutions of Anisotropic Mean Curvature Flow

1 Introduction

2 Preliminaries

2.1 Definitions of Piecewise-Cr Weak Immersion and Its Anisotropic Energy

2.2 Wulff Shape and Convexity of Integrands

3 First Variation Formula, Anisotropic Curvatures, and Anisotropic Gauss Map

4 Outline of the Proof of Theorem 1

5 Examples of Non-convex Integrand and Proofs of Theorems 2 and 3

5.1 A One-Dimensional Example

5.2 Higher Dimensional Examples

5.3 Outline of Proofs of Theorems 2 and 3

6 Anisotropic Mean Curvature Flow

References

The Translating Soliton Equation

1 Historical Introduction and Motivation

2 Examples of Translating Solitons

2.1 Cylindrical Surfaces

2.2 Rotational Surfaces

3 Properties of the Solutions of the Translating Soliton Equation

4 The Dirichlet Problem in Bounded Convex Domains

5 The Dirichlet Problem for the Constant Weighted Mean Curvature

6 The Dirichlet Problem in Unbounded Domains

References

Constant Mean Curvature Surfaces for the Bessel Equation

1 The Generalised Weierstrass Representation

2 The Bessel Equation

3 The zAP Lemma and Perturbations at a Simple Pole

4 Constructing CMC Cylinders

4.1 Cylinder Potential

4.2 Local Gauge

4.3 Unitary Monodromy on the Twice-Puncture Riemann Sphere

4.4 Main Theorem

5 Symmetry

5.1 Cylinders with One Reflection

References

Minimal Surfaces Under Constrained Willmore Transformation

1 Introduction

2 The Willmore Energy

3 Conformal Invariance and the Central Sphere Congruence

4 Constrained Willmore Surfaces and Harmonicity

5 Isothermic Constrained Willmore Surfaces

6 Transformations of Constrained Willmore Surfaces

6.1 Spectral Deformation

6.2 Bäcklund Transformation

7 Polynomial Conserved Quantities for Constrained Willmore Surfaces

8 Transformations of Special Constrained Willmore Surfaces

9 Constant Mean Curvature Surfaces Under Constrained Willmore Transformation

References

On an Enneper–Weierstrass-Type Representation of Constant Gaussian Curvature Surfaces in 3-Dimensional Hyperbolic Space

1 Introduction

1.1 Background

1.2 Main Result

1.3 Labourie’s Compactness Theorem

1.4 General Properties of k-Surfaces

1.5 Limit Points of the Immersion

1.6 Limit Points of the Weierstrass Map

1.7 Systoles

References

Loss of Initial Data Under Limits of Ricci Flows

1 Introduction

2 The Construction

References

Semi-discrete Maximal Surfaces with Singularities in Minkowski Space

1 Introduction

1.1 Overview

1.2 Previous Works

1.3 Main Results

2 Semi-discrete Isothermic Surfaces in mathbbR2,1

3 Proof of Theorem 1

4 Examples of Semi-discrete Isothermic Maximal Surfaces

5 Associated Families of Semi-discrete Maximal Surfaces

6 Singularities of Semi-discrete Maximal Surfaces

6.1 Singularities of Semi-discrete Isothermic Maximal Surfaces

6.2 Singularities of Associated Families of Semi-discrete Isothermic Maximal Surfaces

References

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