Semi-Discrete Isothermic Surfaces

We study mappings of the form ${x : \mathbb{Z}\times\mathbb{R}\to\mathbb{R}^3}$ which can be seen as a limit case of purely discrete surfaces, or as a semi-discretization of smooth surfaces. In particular we discuss circular surfaces, isothermic surfaces, conformal mappings, and dualizability in the sense of Christoffel. We arrive at a semi-discrete version of Koenigs nets and show that in the setting of circular surfaces, isothermicity is the same as dualizability. We show that minimal surfaces constructed as a dual of a sphere have vanishing mean curvature in a certain well-defined sense, and we also give an incidence-geometric characterization of isothermic surfaces.     

Bonnet Ruled Surfaces

We consider the Bonnet ruled surfaces which admit only one non-trivial isometry that preserves the principal curvatures. We determine the Bonnet ruled surfaces whose generators and orthogonal trajectories form a special net called an A-net.     

Bonnet Surfaces with Constant Curvature

We classify Bonnet surfaces with constant curvature in 3-dimensional space forms and show that they are parametrized by curves in 2-dimensional space forms with specific geodesic curvature.     

Isothermic and S-Willmore Surfaces as Solutions to a Problem of Blaschke

We consider the generalization of a classical problem of Blaschke to the higher codimensional case, characterizing Darboux pairs of isothermic surfaces and dual S-Willmore surfaces as the only non-trivial surface pairs that envelop a 2-sphere congruence and conformally correspond to each other. When the sphere congruence consists of the mean curvature spheres of one enveloping surface f, f must be a CMC-1 surface in hyperbolic 3-space, or an S-Willmore surface.     

Isothermic Submanifolds

We propose an answer to a question raised by F. Burstall: Is there any interesting theory of isothermic submanifolds of ? ⁿ of dimension greater than two? We call an n-immersion f(x) in ? ^m isothermic _k if the normal bundle of f is flat and x is a line of curvature coordinate system such that its induced metric is of the form $\sum_{i=1}^{n} g_{ii}\,\mathrm{d} x_{i}^{2}$ with $\sum_{i=1}^{n} \epsilon_{i} g_{ii}=0$ , where ?? _i =1 for 1??i??n?k and ?? _i =?1 for n?k<i??n. A smooth map (f ₁,??,f _n ) from an open subset ${\mathcal{O}}$ of ? ⁿ to the space of m×n matrices is called an n-tuple of isothermic _k n-submanifolds in ? ^m if each f _i is an isothermic _k immersion, $(f_{i})_{x_{j}}$ is parallel to $(f_{1})_{x_{j}}$ for all 1??i,j??n, and there exists an orthonormal frame (e ₁,??,e _n ) and a GL(n)-valued map (a _ij ) such that $\mathrm{d}f_{i}= \sum_{j=1}^{n} a_{ij} e_{j}\,\mathrm {d} x_{j}$ for 1??i??n. Isothermic₁ surfaces in ?³ are the classical isothermic surfaces in ?³. Isothermic _k submanifolds in ? ^m are invariant under conformal transformations. We show that the equation for n-tuples of isothermic _k n-submanifolds in ? ^m is the $\frac{O(m+n-k,k)}{O(m)\times O(n-k,k)}$ -system, which is an integrable system. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries.     

Isothermic Surfaces in Spaces of Arbitrary Dimension: Integrability, Discretization, and Bäcklund Transformations—A Discrete Calapso Equation

A vector analog of the classical Calapso equation governing isothermic surfaces in R ^{n +2} is introduced. It is shown that this vector Calapso system admits a nonlocal) scalar Lax pair based on the classical Moutard equation. The analog of Darboux's Bäcklund transformation for isothermic surfaces in R³ is derived in a systematic manner and shown that it may be formulated in terms of the classical Moutard transformation acting on the scalar Lax pair. A permutability theorem for isothermic surfaces is set down that manifests itself in an explicit superposition principle for the vector Calapso system. This superposition principle in vectorial form is shown to constitute an integrable discretization of the vector Calapso system and, therefore, defines discrete isothermic surfaces in R ^{n +2}. The discrete Calapso equation is related to the discrete Korteweg–de Vries equation and discrete holomorphic functions. A matrix Lax pair based on Clifford algebras and a scalar Lax pair are derived for the discrete Calapso equation. A discrete Moutard-type transformation for the discrete Calapso equation is obtained, and it is shown that the discrete Calapso equation may be specialized to an integrable discrete version of the O( n +2) nonlinear σ-model.     

Isothermic constrained Willmore tori in 3-space

Annals of Global Analysis and Geometry - We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore...     

On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

Ringed surfaces and canal surfaces are surfaces that contain a one-parameter family of circles. Ringed surfaces can be described by a radius function, a directrix curve and vector field along the directrix curve, which specifies the normals of the planes that contain the circles. In particular, the class of ringed surfaces includes canal surfaces, which can be obtained as the envelopes of a one-parameter family of spheres. Consequently, canal surfaces can be described by a spine curve and a radius function. We present parameterization algorithms for rational ringed surfaces and rational canal surfaces. It is shown that these algorithms may generate any rational parameterization of a ringed (or canal) surface with the property that one family of parameter lines consists of circles. These algorithms are used to obtain rational parameterizations for Darboux cyclides and to construct blends between pairs of canal surfaces and pairs of ringed surfaces.     

Orthogonal Surfaces and Their CP-Orders

Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and three-polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which non-generic orthogonal surfaces have a polytopal structure. We review the state of knowledge of the three-dimensional situation. On that basis we introduce terminology for higher dimensional orthogonal surfaces and continue with the study of characteristic points and the cp-orders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cp-orders can lack key properties of face lattices. We investigate extra requirements which may help to have cp-orders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces.      

Schnyder Woods and Orthogonal Surfaces

In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the Brightwell-Trotter Theorem which says, that the face lattice of a 3-polytope minus one face has order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice. Coplanar orthogonal surfaces are in correspondence with a large class of convex straight line drawings of 3-connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time.     

Hyperelliptic Surfaces and their Moduli

The hyperelliptic portion of the moduli space of compact Riemann surfaces of genus g2 is decomposed into a lattice of nondisjoint subvarieties corresponding precisely with the lattice of maximal g-hyperelliptic group actions (classified up to topological equivalence). The resulting stratification of the hyperelliptic moduli space exhibits regularities which depend on the parity of g and can be detected at the level of groups of order 8.     

Unknotting Tunnels and Seifert Surfaces

Let K be a knot with an unknotting tunnel and suppose thatK is not a 2-bridge knot. There is an invariant = p/q Q/2Z,with p odd, defined for the pair (K, ). The invariant has interesting geometric properties. It is oftenstraightforward to calculate; for example, for K a torus knotand an annulus-spanning arc, (K, ) = 1. Although is definedabstractly, it is naturally revealed when K is put in thinposition. If 1 then there is a minimal-genus Seifert surfaceF for K such that the tunnel can be slid and isotoped to lieon F. One consequence is that if (K, ) 1 then K > 1. Thisconfirms a conjecture of Goda and Teragaito for pairs (K, )with (K, ) 1. 2000 Mathematics Subject Classification 57M25,57M27.     

Numerical Burniat and Irregular Surfaces

In this paper, we construct three numerical Burniat surfaces as desingularizations of double planes of degree >10. Two are surfaces having the bigenus P ₂=4 and the third is a surface having the bigenus P ₂=5. In addition, another surface of general type is constructed as a desingularization of a double plane of degree 12 having the birational invariants: q=p _g =1, P ₂=4. One of the numerical Burniat surfaces with P ₂=4 is obtained as a desingularization of a double plane of degree 22 with an irreducible branch locus, so it is a good candidate for having torsion zero. Moreover, its bicanonical transformation seems to be birational.     

Pseudospherical Surfaces and Evolution Equations

We consider evolution equations, mainly of type u_t = F(u, u_x,..., ?^ku/?x^k), which describe pseudo-spherical surfaces. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. Moreover, we investigate how the geometrical properties of surfaces provide analytic information for such equations.