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Register a new userIsothermic surfaces in sphere geometries as Moutard nets
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We give an elaborated treatment of discrete isothermic surfaces and their analogs in different geometries (projective, Mobius, Laguerre, Lie). We find the core of the theory to be a novel projective characterization of discrete isothermic nets as Moutard nets. The latter belong to projective geometry and are nets with planar faces defined through a five-point property: a vertex and its four diagonal neighbors span a three dimensional space. Analytically this property is equivalent to the existence of representatives in the space of homogeneous coordinates satisfying the discrete Moutard equation. Restricting the projective theory to quadrics, we obtain Moutard nets in sphere geometries. In particular, Moutard nets in Mobius geometry are shown to coincide with discrete isothermic nets. The five-point property in this particular case says that a vertex and its four diagonal neighbors lie on a common sphere, which is a novel characterization of discrete isothermic surfaces. Discrete Laguerre isothermic surfaces are defined through the corresponding five-plane property which requires that a plane and its four diagonal neighbors share a common touching sphere. Equivalently, Laguerre isothermic surfaces are characterized by having an isothermic Gauss map. We conclude with Moutard nets in Lie geometry.
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We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios.
A diagonal metric sum_{i=1}^n g_{ii} dx_i^2 is termed Guichard_k if sum_{i=1}^{n-k}g_{ii}-sum_{i=n-k+1}^n g_{ii}=0. A hypersurface in R^{n+1} is isothermic_k if it admits line of curvature co-ordinates such that its induced metric is Guichard_k. Isothermic_1 surfaces in R^3 are the classical isothermic surfaces in R^3. Both isothermic_k hypersurfaces in R^{n+1} and Guichard_k orthogonal co-ordinate systems on R^n are invariant under conformal transformations. A sequence of n isothermic_k hypersurfaces in R^{n+1} (Guichard_k orthogonal co-ordinate systems on R^n resp.) is called a Combescure sequence if the consecutive hypersurfaces (orthogonal co-ordinate systems resp.) are related by Combescure transformations. We give a correspondence between Combescure sequences of Guichard_k orthogonal co-ordinate systems on R^n and solutions of the O(2n-k,k)/O(n)xO(n-k,k)-system, and a correspondence between Combescure sequences of isothermic_k hypersurfaces in R^{n+1} and solutions of the O(2n+1-k,k)/O(n+1)xO(n-k,k)-system, both being integrable systems. 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.
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.
We investigate the geometry of almost Robinson manifolds, Lorentzian analogues of Hermitian manifolds, defined by Nurowski and Trautman as Lorentzian manifolds of even dimension equipped with a totally null complex distribution of maximal rank. Associated to such a structure, there is a congruence of null curves, which, in dimension four, is geodesic and non-shearing if and only if the complex distribution is involutive. Under suitable conditions, the distribution gives rise to an almost Cauchy--Riemann structure on the leaf space of the congruence. We give a comprehensive classification of such manifolds on the basis of their intrinsic torsion. This includes an investigation of the relation between an almost Robinson structure and the geometric properties of the leaf space of its congruence. We also obtain conformally invariant properties of such a structure, and we finally study an analogue of so-called generalised optical geometries as introduced by Robinson and Trautman.
We present a multi-scale model to study the attachment of spherical particles with a rigid core, coated with binding ligands and in equilibrium with the surrounding, quiescent fluid medium. This class of fluid-immersed adhesion is widespread in many natural and engineering settings. Our theory highlights how the micro-scale binding kinetics of these ligands, as well as the attractive / repulsive surface potential in an ionic medium effects the eventual macro-scale size distribution of the particle aggregates (flocs). The results suggest that the presence of elastic ligands on the particle surface allow large floc aggregates by inducing efficient inter-floc collisions (i.e., a large, non-zero collision factor). Strong electrolytic composition of the surrounding fluid favors large floc formation as well.
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