Quaternionic holomorphic geometry: Pluecker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean curvature tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue estimates on Riemannian spin bundles in dimension 2 are given.Comment: 70 pages, 1 figur

Periodic discrete conformal maps

A discrete conformal map (DCM) maps the square lattice to the Riemann sphere such that the image of every irreducible square has the same cross-ratio. This paper shows that every periodic DCM can be determined from spectral data (a hyperelliptic compact Riemann surface, called the spectral curve, equipped with some marked points). Each point of the map corresponds to a line bundle over the spectral curve so that the map corresponds to a discrete subgroup of the Jacobi variety. We derive an explicit formula for the generic maps using Riemann theta functions, describe the typical singularities and give a geometric interpretation of DCM's as a discrete version of the Schwarzian KdV equation. As such, the DCM equation is a discrete soliton equation and we describe the dressing action of a loop group on the set of DCM's. We also show that this action corresponds to a lattice of isospectral Darboux transforms for the finite gap solutions of the KdV equation.Comment: 41 pages, 10 figures, LaTeX2

Bonnet pairs and isothermic surfaces

In this note we classify all Bonnet pairs on a simply connected domain. Our main intent was to apply what we call a quaternionic function theory to a concrete problem in differential geometry. The ideas are simple: conformal immersions into quaternions or imaginary quaternions take the place of chart maps for a Riemann surface. Starting from a reference immersion we construct all conformal immersions of a given (simply connected) Riemann surface (up to translational periods) by spin transformations. With this viewpoint in mind we discuss how to construct all Bonnet pairs on a simply connected domain from isothermic surfaces and vice versa. Isothermic surfaces are solutions to a certain soliton equation and thus a simple dimension count tells us that we obtain Bonnet pairs which are not part of any of the classical Bonnet families. The corresponcence between Bonnet pairs and isothermic surfaces is explicit and to each isothermic surface we obtain a 4-parameter family of Bonnet pairs

Isothermic surfaces and conservation laws

For CMC surfaces in $3$-dimensional space forms, we relate the moment class of Korevaar--Kusner--Solomon to a second cohomology class arising from the integrable systems theory of isothermic surfaces. In addition, we show that both classes have a variational origin as Noether currents

Isothermic submanifolds of symmetric RR-spaces

We extend the classical theory of isothermic surfaces in conformal 3-space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric $R$-spaces with essentially no loss of integrable structure.Comment: 35 pages, 3 figures. v2: typos and other infelicities corrected

Conformal Geometry of Surfaces in S4 and Quaternions

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