In the present work, a coordinate-free way is suggested to handle conformal maps of a Rie­mannian sur­face into a space of constant curvature of maximum dimension 4, modeled on the non-commutative field of quaternions. This setup for the target space and the idea to treat dif­fe­rential 2-forms on Rie­mannian surfaces as quadratic functions on the tangent space, are the starting points for the development of the theory of conformal maps and in particular of con­formal immersions. As a first result, very nice condi­tions for the conformality of immersions into 3- and 4-dimensional space-forms are deduced and a sim­ple way to write the second fun­damental form is found. If the target space is euclidean 3-space, an alternative approach is proposed by fixing a spin structure on the Rie­mannian surface. The problem of finding a local immersion is then reduced to that of solving a linear Dirac equation with a potential whose square is the Willmore in­tegrand. This allows to make statements about the structure of the moduli space of conformal immersions and to derive a very nice criterion for a conformal immersion to be con­strained Willmore. As an application the Dirac equation with constant potential over spheres and tori is solved. This yields explicit immersion formulae out of which there were produced pictures, the Dirac-spheres and -tori. These immersions have the property that their Willmore integrand generates a metric of vanishing and constant curvature, respectively. As a next step an affine immersion theory is developped. This means, one starts with a given conformal immersion into euclidean 3-space and looks for new ones in the same conformal class. This is called a spin-transformation and it leads one to solve an affine Dirac equation. Also, it is shown how the coordi­nate-dependent generalized Weierstrass representation fits into the present framework. In particular, it is now natural to consider the class of conformal im­mersions that admit new conformal immersions having the same potential. It turns out, that all geometri­cally interesting immersions admit such an isopotential spin-transformation and that this property of an immersion is even a conformal invariant of the ambient space. It is shown that conformal isothermal immersions generate both via their dual and via Darboux trans­formations non-trivial families of new isopotential conformal immersions. Similarly to this, conformal (constrained) Willmore immersions produce non-trivial families of isopotential im­mer­sions of which subfamilies are (constrained) Willmore again having even the same Will­more integral. Another obser­vation is, that the Euler-Lagrange equation for the Willmore pro­blem is the integrability condition for a quaternionic 1-form, which generates a conformal mi­nimal im­mersions into hyperbolic 4-space. Vice versa, any such immersion determines a con­formal Willmore immersion. As a conse­quence, there is a one-to-one correspondence between con­formal minimal immersions into Lorentzian space and those into hyperbolic space, which gene­ralizes to any dimension. There is also induced an action on conformal minimal immersi­ons into hyperbolic 4-space. Another fact is, that conformal con­stant mean curvature (cmc) immersions into some 3-dimensional space form unveil to be isothermal and constrained Will­more. The reverse statement is true at least for tori. Finally a very simple proof of a theorem by R.Bryant concer­ning Willmore spheres is given. In the last part, time-dependent conformal immersions are considered. Their deformation for­mulae are computed and it is investigated under what conditions the flow commutes with Moe­bius transforma­tions. The modified Novikov-Veselov flow is written down in a conformal in­variant way and explicit deformation formulae for the immersion function itself and all of its invariants are given. This flow commutes with Moebius transformations. Its definition is cou­pled with a delta-bar problem, for which a so­lution is presented under special conditions. These are fulfilled at least by cmc immersions and by sur­faces of revolution and the general flow for­mulae reduce to very nice formulae in these cases.

The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.

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.

Contents:Affine Bibliography 1998 (T Binder et al.)Contact Metric R-Harmonic Manifolds (K Arslan & C Murathan)Local Classification of Centroaffine Tchebychev Surfaces with Constant Curvature Metric (T Binder)Hypersurfaces in Space Forms with Some Constant Curvature Functions (F Brito et al.)Some Relations Between a Submanifold and Its Focal Set (S Carter & A West)On Manifolds of Pseudosymmetric Type (F Defever et al.)Hypersurfaces with Pseudosymmetric Weyl Tensor in Conformally Flat Manifolds (R Deszcz et al.)Least-Squares Geometrical Fitting and Minimising Functions on Submanifolds (F Dillen et al.)Cubic Forms Generated by Functions on Projectively Flat Spaces (J Leder)Distinguished Submanifolds of a Sasakian Manifold (I Mihai)On the Curvature of Left Invariant Locally Conformally Para-Kählerian Metrics (Z Olszak)Remarks on Affine Variations on the Ellipsoid (M Wiehe)Dirac's Equation, Schrödinger's Equation and the Geometry of Surfaces (T J Willmore)and other papers Readership: Researchers doing differential geometry and topology. Keywords:Proceedings;Geometry;Topology;Valenciennes (France);Lyon (France);Leuven (Belgium);Dedication

* Invited articles in differential geometry and mathematical physics in honor of Hideki Omori * Focus on recent trends and future directions in symplectic and Poisson geometry, global analysis, Lie group theory, quantizations and noncommutative geometry, as well as applications of PDEs and variational methods to geometry * Will appeal to graduate students in mathematics and quantum mechanics; also a reference

This book is the first monograph dedicated entirely to Willmore energy and Willmore surfaces as contemporary topics in differential geometry. While it focuses on Willmore energy and related conjectures, it also sits at the intersection between integrable systems, harmonic maps, Lie groups, calculus of variations, geometric analysis and applied differential geometry. Rather than reproducing published results, it presents new directions, developments and open problems. It addresses questions like: What is new in Willmore theory? Are there any new Willmore conjectures and open problems? What are the contemporary applications of Willmore surfaces? As well as mathematicians and physicists, this book is a useful tool for postdoctoral researchers and advanced graduate students working in this area.

This book introduces the reader to the geometry of surfaces and submanifolds in the conformal n-sphere.