Constrained Willmore Surfaces: Symmetries of a Möbius Invariant Integrable System

Áurea Casinhas Quintino

doi:10.1017/9781108885478

From Bäcklund to Darboux, this monograph presents a comprehensive journey through the transformation theory of constrained Willmore surfaces, a topic of great importance in modern differential geometry and, in particular, in the field of integrable systems in Riemannian geometry. The first book on this topic, it discusses in detail a spectral deformation, Bäcklund transformations and Darboux transformations, and proves that all these transformations preserve the existence of 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, and bridging the gap between different approaches to the subject, classical and modern. Clearly written with extensive references, chapter introductions and self-contained accounts of the core topics, it is suitable for newcomers to the theory of constrained Wilmore surfaces. Many detailed computations and new results unavailable elsewhere in the literature make it also an appealing reference for experts.

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Contents

Frontmatter
pp i-iv
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Dedication
pp v-vi
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Contents
pp vii-x
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Preface
pp xi-xiv
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Introduction
pp 1-15
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1 - A Bundle Approach to Conformal Surfaces in Space-Forms
pp 16-30
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2 - The Mean Curvature Sphere Congruence
pp 31-44
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3 - Surfaces under Change of Flat Metric Connection
pp 45-48
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4 - Willmore Surfaces
pp 49-72
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5 - The Euler–Lagrange ConstrainedWillmore Surface Equation
pp 73-86
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6 - Transformations of Generalized Harmonic Bundles and Constrained Willmore Surfaces
pp 87-134
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7 - Constrained Willmore Surfaces with a Conserved Quantity
pp 135-144
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8 - Constrained Willmore Surfaces and the Isothermic Surface Condition
pp 145-182
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9 - The Special Case of Surfaces in 4-Space
pp 183-234
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Appendix A - Hopf Differential and Umbilics
pp 235-236
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Appendix B - Twisted vs. Untwisted Bäcklund Transformation Parameters
pp 237-239
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References
pp 240-244
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Index
pp 245-246
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Constrained Willmore Surfaces

Symmetries of a Möbius Invariant Integrable System

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Contents

Frontmatter
pp i-iv

Dedication
pp v-vi

Contents
pp vii-x

Preface
pp xi-xiv

Introduction
pp 1-15

1 - A Bundle Approach to Conformal Surfaces in Space-Forms
pp 16-30

2 - The Mean Curvature Sphere Congruence
pp 31-44

3 - Surfaces under Change of Flat Metric Connection
pp 45-48

4 - Willmore Surfaces
pp 49-72

5 - The Euler–Lagrange ConstrainedWillmore Surface Equation
pp 73-86

6 - Transformations of Generalized Harmonic Bundles and Constrained Willmore Surfaces
pp 87-134

7 - Constrained Willmore Surfaces with a Conserved Quantity
pp 135-144

8 - Constrained Willmore Surfaces and the Isothermic Surface Condition
pp 145-182

9 - The Special Case of Surfaces in 4-Space
pp 183-234

Appendix A - Hopf Differential and Umbilics
pp 235-236

Appendix B - Twisted vs. Untwisted Bäcklund Transformation Parameters
pp 237-239

References
pp 240-244

Index
pp 245-246

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Constrained Willmore Surfaces

Symmetries of a Möbius Invariant Integrable System

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