Constrained Willmore Surfaces (London Mathematical Society Lecture Note Series, Series Number 465)

Front matter
Copyright
Dedications
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 Surtaces 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 ∧2ℝn+1,1≅o(ℝn+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 Surtaces
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 Backlund Transformation of Constrained Harmonic Bundles and Constrained Willmore Surfaces
	6.12 Real Backlund Transformation of Constrained Harmonic Bundles and Constrained Willmore Surfaces
	6.13 Spectral Deformation vs. Backlund Transformation
7 Constrained Willmore Surfaces with a Conserved Quantity
	7.1 Conserved Quanities 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 Transtormation
		7.3.1 Conserved Quantities under Constrained Willmore Spectral Deformation
		7.3.2 Conscrved Quantitics under Constrained Willmore Backlund Transformation
8 Constrained Willmore Surfaces and the Isothermic Surface
	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 Surtaces 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 Spectra Deformations
		8.2.4 CMC Surfaces under Constrained Willmore Backlund 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 S4 ≅ ℍP1
		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 Backlund Transformation of Constrained Willmore Surfaces in 4-Space
		9.3.2 Twisted vs. Untwisted Backlund Transformation of Constrained Willmore Surfaces in 4-Space
		9.3.3 Darboux Transformation of Constrained Willmore Surfaces in 4-Space
		9.3.4 Backlund Transformation vs. Darboux Transformation of Constrained Willmore Surfaces in 4-Space
Appendix A Hopf Differential and Umbilics
Appendix B Twisted vs. Untwisted Backlund Transformation Parameters
References
Index