Constrained Willmore surfaces symmetries of a Möbius invariant integrable system Áurea Casinhas Quintino

Format:

Book

Author/Creator:

Quintino, Áurea Casinhas, 1974- author.

Series:

London Mathematical Society lecture note series 465

London mathematical society lecture note series 465

Language:

English

Subjects (All):

Transformations (Mathematics).

Möbius transformations.

Physical Description:

1 online resource

Place of Publication:

Cambridge, United Kingdom New York, NY Cambridge University Press 2021

Summary:

"This work is dedicated to the study of the Möbius invariant class of constrained Willmore surfaces and its symmetries. Characterized by the perturbed harmonicity of the mean curvature sphere congruence, a generalization of the well-developed integrable systems theory of harmonic maps emerges. The starting point is a zero-curvature characterization, due to Burstall-Calderbank, which we derive from the underlying variational problem. Constrained Willmore surfaces come equipped with a family of flat metric connections. We then define a spectral deformation, by the action of a loop of flat metric connections; Bäcklund transformations, defined by the application of a version of the Terng-Uhlenbeck dressing action by simple factors; and, in 4-space, Darboux transformations, based on the solution of a Riccati equation, generalizing the transformation of Willmore surfaces presented in the quaternionic setting by Burstall-Ferus-Leschke-Pedit-Pinkall. We establish a permutability between spectral deformation and Bäcklund transformation and prove that non-trivial Darboux transformation of constrained Willmore surfaces in 4-space can be obtained as a particular case of Bäcklund transformation. All these transformations corresponding to the zero Lagrange multiplier preserve the class of Willmore surfaces. We verify that both spectral deformation and Bäcklund transformation preserve the class of constrained Willmore surfaces admitting a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, in the latter case. Constrained Willmore transformation proves to be unifying to the rich transformation theory of CMC surfaces in 3-space"-- Provided by publisher

Contents:

Cover

Series information

Title page

Copyright information

Dedication

Contents

Preface

Introduction

1 A Bundle Approach to Conformal Surfaces in Space-Forms

1.1 Space-Forms in the Conformal Projectivized Light Cone

1.2 Conformal Surfaces in the Light Cone Picture

1.2.1 Oriented Conformal Surfaces: Generalities

1.2.2 Conformal Immersions of Surfaces into the Projectivized Light Cone

2 The Mean Curvature Sphere Congruence

2.1 Mean Curvature and Central Sphere Congruence

2.2 The Normal Bundle to the Central Sphere Congruence

2.3 Conformal Gauss Map and Gauss-Codazzi-Ricci Equations

2.3.1 The Exterior Power ∧[sup(2)]R[sup(n+1,1)] [cong] o(R[sup(n+1,1)])

2.3.2 The Gauss-Ricci and Codazzi Equations

3 Surfaces under Change of Flat Metric Connection

4 Willmore Surfaces

4.1 The Willmore Functional

4.2 Willmore Surfaces: Definition

4.3 Willmore Energy vs. Dirichlet Energy

4.4 Willmore Surfaces and Harmonicity

4.5 The Euler-Lagrange Willmore Surface Equation

4.6 Willmore Surfaces under Change of Flat Metric Connection

4.7 Spectral Deformation of Willmore Surfaces

5 The Euler-Lagrange Constrained Willmore Surface Equation

5.1 Constrained Willmore Surfaces: Definition

5.2 The Hopf Differential and the Schwarzian Derivative

5.3 The Euler-Lagrange Constrained Willmore Surface Equation

5.4 Constrained Willmore Surfaces: An Equation on the Hopf Differential and the Schwarzian Derivative

6 Transformations of Generalized Harmonic Bundles and Constrained Willmore Surfaces

6.1 Constrained Harmonic Bundles

6.2 Constrained Harmonicity: A Zero-Curvature Characterization

6.3 Constrained Willmore Surfaces and Constrained Harmonicity

6.4 Constrained Willmore Surfaces: A Zero-Curvature Characterization

6.5 Spectral Deformation of Constrained Harmonic Bundles

6.6 Complexified Constrained Willmore Surfaces

6.7 Constrained Willmore Surfaces under Change of Flat Metric Connection

6.8 Spectral Deformation of Constrained Willmore Surfaces

6.9 Real Spectral Deformation of Constrained Willmore Surfaces

6.10 Dressing Action

6.11 Bäcklund Transformation of Constrained Harmonic Bundles and Constrained Willmore Surfaces

6.12 Real Bäcklund Transformation of Constrained Harmonic Bundles and Constrained Willmore Surfaces

6.13 Spectral Deformation vs. Bäcklund Transformation

7 Constrained Willmore Surfaces with a Conserved Quantity

7.1 Conserved Quantities of Constrained Willmore Surfaces

7.2 Constrained Willmore Surfaces with a Conserved Quantity: Examples

7.2.1 The Special Case of Codimension 1: CMC surfaces in 3-Dimensional Space-Forms

7.2.2 A Special Case in Codimension 2: holomorphic mean curvature vector surfaces in 4-Dimensional Space-Forms

7.3 Conserved Quantities under Constrained Willmore Transformation

7.3.1 Conserved Quantities under Constrained Willmore Spectral Deformation

7.3.2 Conserved Quantities under Constrained Willmore Bäcklund Transformation

8 Constrained Willmore Surfaces and the Isothermic Surface Condition

8.1 Isothermic Surfaces

8.1.1 Isothermic Surfaces: Definition

8.1.2 Isothermic Surfaces and Hopf Differential

8.1.3 Isothermic Surfaces: A Zero-Curvature Characterization

8.1.4 Transformations of Isothermic Surfaces

8.1.5 Isothermic Surfaces under Constrained Willmore Transformation

8.1.6 Isothermic Surface Condition and Uniqueness of Multiplier

8.2 Constant Mean Curvature Surfaces in 3-Dimensional Space-Forms

8.2.1 CMC Surfaces as Isothermic Constrained Willmore Surfaces with a Conserved Quantity

8.2.2 CMC Surfaces: An Equation on the Hopf Differential and the Schwarzian Derivative

8.2.3 CMC Surfaces at the Intersection of Spectral Deformations

8.2.4 CMC Surfaces under Constrained Willmore Bäcklund Transformation

8.2.5 CMC Surfaces at the Intersection of Integrable Geometries

9 The Special Case of Surfaces in 4-Space

9.1 Surfaces in S[sup(4)] [cong] HP[sup(1)]

9.1.1 Linear Algebra

9.1.2 The Mean Curvature Sphere Congruence

9.1.3 Mean Curvature Sphere Congruence and Central Sphere Congruence

9.2 Constrained Willmore Surfaces in 4-Space

9.3 Transformations of Constrained Willmore Surfaces in 4-Space

9.3.1 Untwisted Bäcklund Transformation of Constrained Willmore Surfaces in 4-Space

9.3.2 Twisted vs. Untwisted Bäcklund Transformation of Constrained Willmore Surfaces in 4-Space

9.3.3 Darboux Transformation of Constrained WillmoreSurfaces in 4-Space

9.3.4 Bäcklund Transformation vs. Darboux Transformation of Constrained Willmore Surfaces in 4-Space

Appendix A Hopf Differential and Umbilics

Appendix B Twisted vs. Untwisted Bäcklund Transformation Parameters

References

Index

Notes:

Includes bibliographical references and index

Online resource; title from digital title page (viewed on May 21, 2021)

Other Format:

Print version Quintino, Áurea Casinhas, 1974- Constrained Willmore surfaces

ISBN:

9781108885478

1108885470

9781108882200

110888220X

OCLC:

1198087974

Access Restriction:

Restricted for use by site license