Constrained Willmore Surfaces

Symmetries of a Möbius Invariant Integrable System

Áurea Casinhas Quintino

• Publisher: Cambridge University Press
• Year: 2021
• Print ISBN: 9781108794428
• eBook ISBN: 9781108882200
• Topic: Matematica
• Subtopic: Topologia

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.

Table of contents

• Cover
• Series information
• Title page
• Copyright information
• Dedication
• Contents
• Preface
• Introduction
• 1 A Bundle Approach to Conformal Surfaces in Space-Forms
• 2 The Mean Curvature Sphere Congruence
• 3 Surfaces under Change of Flat Metric Connection
• 4 Willmore Surfaces
• 5 The Euler–Lagrange Constrained Willmore Surface Equation
• 6 Transformations of Generalized Harmonic Bundles and Constrained Willmore Surfaces
• 7 Constrained Willmore Surfaces with a Conserved Quantity
• 8 Constrained Willmore Surfaces and the Isothermic Surface Condition
• 9 The Special Case of Surfaces in 4-Space
• Appendix A Hopf Differential and Umbilics
• Appendix B Twisted vs. Untwisted Bäcklund Transformation Parameters
• References
• Index