60 research outputs found
Constructing solutions to the Bj\"orling problem for isothermic surfaces by structure preserving discretization
In this article, we study an analog of the Bj\"orling problem for isothermic
surfaces (that are more general than minimal surfaces): given a real analytic
curve in , and two analytic non-vanishing orthogonal
vector fields and along , find an isothermic surface that is
tangent to and that has and as principal directions of
curvature. We prove that solutions to that problem can be obtained by
constructing a family of discrete isothermic surfaces (in the sense of Bobenko
and Pinkall) from data that is sampled along , and passing to the limit
of vanishing mesh size. The proof relies on a rephrasing of the
Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its
discretization which is induced from the geometry of discrete isothermic
surfaces. The discrete-to-continuous limit is carried out for the Christoffel
and the Darboux transformations as well.Comment: 29 pages, some figure
Approximation of conformal mappings using conformally equivalent triangular lattices
Consider discrete conformal maps defined on the basis of two conformally
equivalent triangle meshes, that is edge lengths are related by scale factors
associated to the vertices. Given a smooth conformal map , we show that it
can be approximated by such discrete conformal maps . In
particular, let be an infinite regular triangulation of the plane with
congruent triangles and only acute angles (i.e.\ ). We scale this
tiling by and approximate a compact subset of the domain of
with a portion of it. For small enough we prove that there exists a
conformally equivalent triangle mesh whose scale factors are given by
on the boundary. Furthermore we show that the corresponding discrete
conformal maps converge to uniformly in with error of
order .Comment: 14 pages, 3 figures; v2 typos corrected, revised introduction, some
proofs extende
On a new conformal functional for simplicial surfaces
We introduce a smooth quadratic conformal functional and its weighted version
where
is the extrinsic intersection angle of the circumcircles of the
triangles of the mesh sharing the edge and is the valence of
vertex . Besides minimizing the squared local conformal discrete Willmore
energy this functional also minimizes local differences of the angles
. We investigate the minimizers of this functionals for simplicial
spheres and simplicial surfaces of nontrivial topology. Several remarkable
facts are observed. In particular for most of randomly generated simplicial
polyhedra the minimizers of and are inscribed polyhedra. We
demonstrate also some applications in geometry processing, for example, a
conformal deformation of surfaces to the round sphere. A partial theoretical
explanation through quadratic optimization theory of some observed phenomena is
presented.Comment: 14 pages, 8 figures, to appear in the proceedings of "Curves and
Surfaces, 8th International Conference", June 201
On the variational interpretation of the discrete KP equation
We study the variational structure of the discrete Kadomtsev-Petviashvili
(dKP) equation by means of its pluri-Lagrangian formulation. We consider the
dKP equation and its variational formulation on the cubic lattice as well as on the root lattice . We prove that, on a lattice
of dimension at least four, the corresponding Euler-Lagrange equations are
equivalent to the dKP equation.Comment: 24 page
On semidiscrete constant mean curvature surfaces and their associated families
The present paper studies semidiscrete surfaces in three-dimensional Euclidean space within the framework of integrable systems. In particular, we investigate semidiscrete surfaces with constant mean curvature along with their associated families. The notion of mean curvature introduced in this paper is motivated by a recently developed curvature theory for quadrilateral meshes equipped with unit normal vectors at the vertices, and extends previous work on semidiscrete surfaces. In the situation of vanishing mean curvature, the associated families are defined via a Weierstrass representation. For the general cmc case, we introduce a Lax pair representation that directly defines associated families of cmc surfaces, and is connected to a semidiscrete [Formula: see text] -Gordon equation. Utilizing this theory we investigate semidiscrete Delaunay surfaces and their connection to elliptic billiards
Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization
We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in \mathbb {R}^{3}, as hyperelliptic curves, and as \mathbb {CP}^{1} modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller
Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations
A unified framework is presented for the solution structure of three-dimensional discrete integrable systems, including the lattice AKP, BKP and CKP equations. This is done through the so-called direct linearizing transform, which establishes a general class of integral transforms between solutions. As a particular application, novel soliton-type solutions for the lattice CKP equation are obtained
Gross-Neveu Models, Nonlinear Dirac Equations, Surfaces and Strings
Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model
(GN2), and its chiral cousin, the NJL2 model, have shown that there are phases
with inhomogeneous crystalline condensates. These (static) condensates can be
found analytically because the relevant Hartree-Fock and gap equations can be
reduced to the nonlinear Schr\"odinger equation, whose deformations are
governed by the mKdV and AKNS integrable hierarchies, respectively. Recently,
Thies et al have shown that time-dependent Hartree-Fock solutions describing
baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation,
and can be mapped directly to classical string solutions in AdS3. Here we
propose a geometric perspective for this result, based on the generalized
Weierstrass spinor representation for the embedding of 2d surfaces into 3d
spaces, which explains why these well-known integrable systems underlie these
various Gross-Neveu gap equations, and why there should be a connection to
classical string theory solutions. This geometric viewpoint may be useful for
higher dimensional models, where the relevant integrable hierarchies include
the Davey-Stewartson and Novikov-Veselov systems.Comment: 27 pages, 1 figur
Constant mean curvature surfaces in AdS_3
We construct constant mean curvature surfaces of the general finite-gap type
in AdS_3. The special case with zero mean curvature gives minimal surfaces
relevant for the study of Wilson loops and gluon scattering amplitudes in N=4
super Yang-Mills. We also analyze properties of the finite-gap solutions
including asymptotic behavior and the degenerate (soliton) limit, and discuss
possible solutions with null boundaries.Comment: 19 pages, v2: minor corrections, to appear in JHE
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