The paper examines the behavior of Weierstrass measures on stable curves as they approach a nodal stable curve.
problem Understanding the behavior of Weierstrass measures on stable curves as they approach a nodal stable curve.
method Analyzing the limiting behavior of Weierstrass measures on a smooth curve of genus g⩾2 as it approaches a nodal stable curve in the Deligne-Mumford compactification. result The Weierstrass measures on a stable rational curve at the boundary of Mg are completely determined. We give a explicit computation of the pointed harmonic volumes of hyperelliptic curves with Weierstrass base points, which are paraphrased into a combinatorial formula.
The paper counts orbifold points on Teichmüller curves and finds genus bounds.
problem Counting orbifold points on Teichmüller curves and determining their homeomorphism types.
method Analyzing algebraic and geometric properties of Teichmüller curves and using isometric immersions.
result The genus of components of Weierstrass curves WD is determined, and bounds on genus are given. We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve y2=f(x) of arbitrary genus g as solutions of sine-Gordon equation using Weierstrass's local parameters, which are characterized by two ramified points. Though the hyperelliptic solutions of the sine-Gord…
Paper connects surfaces in 4D and 3D spacetime.
problem Finding relations between Lorentz surfaces in different spacetime dimensions.
method Weierstrass-type representations for null curves and surfaces.
result Relation between minimal Lorentz surfaces in R24 and R13. Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.
problem Natural PDEs for minimal Lorentz surfaces in R24. method Weierstrass type representations and canonical coordinates.
result Explicit solution of the system of natural PDEs.
Let W -> A^2 be the universal Weierstrass family of cubic curves over C. For each N >= 2, we construct surfaces parametrizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of A^2. Since W -> A^2 is the versal deformation space of a cusp singul…
Using techniques of integrable systems, we study a Weierstrass representation formula for timelike surfaces with prescribed mean curvature in Minkowski 3-space. It is shown that timelike minimal surfaces are obtained by integrating a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in Minkowski 3-spa…
Transforms Dirac operators to relate surfaces in 4D.
problem Relating surfaces in 4D via Dirac operators.
method Constructs Moutard transformation for 2D Dirac operators.
result Explicit example creates double points on spectral curve.
We construct the spectral curve and the Baker--Akhiezer function for the Dirac operator which corresponds to the Clifford torus via the Weierstrass representation. By constructing this Baker--Akhiezer function we demonstrate a general procedure for constructing Dirac operators and their Baker--Akhiezer functions corres…
This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in n-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret re…
This work extends holomorphic surface representations to isotropic space.
problem Representing minimal surfaces in simply isotropic space with degenerate metrics.
method Developed new forms of Weierstrass and Björling representations for isotropic minimal surfaces.
result Holomorphic representations of isotropic minimal surfaces are achieved.
Paper solves a conjecture about minimal surfaces using sphere intersections and Weierstrass data.
problem Solving the Fraser-Li conjecture for minimal surfaces.
method Using the Weierstrass representation formula and sphere intersections.
result The conjecture can be translated into problems about the Gauss map.
New method constructs translationally equivariant hyperbolic affine spheres.
problem Constructing translationally equivariant hyperbolic affine spheres.
method Noncompact Iwasawa factorization via DPW method and Weierstrass elliptic functions.
result Every translationally equivariant hyperbolic affine sphere is equiaffinely equivalent to one with a circle, hyperbola, or parabola slice curve.
Paper finds explicit expressions for Jenkins-Strebel differentials on a sphere with four poles.
problem Finding explicit Jenkins-Strebel differentials on a Riemann sphere with four poles.
method Using the Weierstrass ℘ function and simple closed curves. result Explicit expressions and algorithm for Jenkins-Strebel differentials on the sphere with four poles.
Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in real, complex and quaternionic projective spaces or complex quadrics. The paper ge…
Algorithms compute the topology of hyperelliptic curves in 2D and 3D.
problem Computing the topology of hyperelliptic curves in higher dimensions.
method Birational mapping of the plane or space to compute the topology of the curve.
result Algorithms implemented in { t Maple} for computing the topology of hyperelliptic curves.
Discrete approximation solves Björling's minimal surface problem.
problem Constructing minimal surfaces from real-analytic curves with specified normal fields.
method Approximate solution by discrete minimal surfaces and discrete isothermic surfaces.
result Approximation error is proportional to the square of the mesh size.
Paper estimates Gaussian curvature of minimal graphs in a specific manifold.
problem Estimating Gaussian curvature of minimal graphs in MimesR. method Using Weierstrass representation via ℘−harmonic mappings and Schwarz lemma type results. result Proves Schwarz lemma type and Heinz type results for harmonic mappings.
New algebraic parametrizations for harmonic maps to orthogonal group.
problem Finding solutions to harmonic maps from surfaces to O(n).
method Algebraic parametrizations and free holomorphic data.
result Reveals correspondence with null curves and minimal surfaces.
Data science reveals patterns in elliptic curve ranks and coefficients.
problem Understanding rational points on elliptic curves via BSD conjecture.
method Data science, machine learning, topological data analysis.
result Patterns and distributions in rank versus Weierstrass coefficients.
We solve the analogue of Björling's problem for Willmore surfaces via a harmonic map representation. For the umbilic-free case the problem and solution are as follows: given a real analytic curve y0 in S3, together with the prescription of the values of the surface normal and the dual Willmore surface along the c…
This paper studies non-compact Ricci surfaces with catenoidal ends.
problem Characterizing non-compact Ricci surfaces with catenoidal ends.
method Using Weierstrass data and minimal immersion techniques.
result Classification and existence results for positive genus Ricci surfaces.
In the previous work (J. Geom. Phys. {\bf{39}} (2001) 50-61), the closed loop solitons in a plane, \it i.e., loops whose curvatures obey the modified Korteweg-de Vries equations, were investigated for the case related to algebraic curves with genera one and two. This article is a generalization of the previous article …
Minimal cylinders in Heisenberg group characterized using loop group method.
problem Characterizing minimal cylinders in the Heisenberg group.
method Generalized Weierstrass type representation and loop group method.
result Characterization of all non-vertical minimal cylinders in terms of pairs of closed plane curves.
Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only …
We show that any equation from the Davey--Stewartson hierarchy induces an infinite family of geometrically different deformations of tori in R4 preserving the Willmore functional. We expose a derivation of the Weierstrass representation for surfaces in the four-space which is not unique in difference from the case …
This paper establishes an interesting connection between the family of CMC surfaces of revolution in E13 and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in the E13, only spacelike cylinders and stand…
In this paper we consider two special classes of constrained Willmore tori in the 3-sphere. The first class is given by the rotation of closed elastic curves in the upper half plane - viewed as the hyperbolic plane - around the x-axis. The second is given as the preimage of closed constrained elastic curves, i.e., elas…
A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvat…
New representations for discrete surfaces derived from dual transforms.
problem Constructing discrete surfaces in differential geometry.
method Using Ω-dual transform and lightlike Gauss maps in Laguerre geometry. result All discrete linear Weingarten surfaces arise via Weierstrass-type representations.
Unified description of Weierstrass-type representations.
problem Creating surfaces with special curvature properties.
method Unified description using classical transformation theory of Ω-surfaces.
result Unified understanding of Weierstrass-type representations.
The paper solves a mean field equation on hyperelliptic curves and examines its diabatic limit.
problem Solving a mean field equation on hyperelliptic curves.
method Algebraic construction and study of rescaled equation with adiabatic limit.
result An algebraic solution to the mean field equation on hyperelliptic curves and its diabatic limit.
Connected Prym eigenforms found in a specific mathematical space.
problem Understanding the structure of Prym eigenforms in a particular mathematical space.
method Proving the connectedness of Prym eigenform loci and classifying square-tiled surfaces.
result The projection of Prym eigenform loci is a single Teichmüller curve.
Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smootha…
Lecture notes on Lorentz geometry, focusing on curves and surfaces.
problem Diagonalization of the Weingarten map for timelike surfaces and classification of surfaces with constant Gaussian curvature.
method Analysis of linear algebra in pseudo-Euclidean space, application of the Fundamental Theorem of Curves, and use of split-complex algebra.
result Local classification of surfaces with constant Gaussian curvature and Weierstrass' representation formula.
A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian…
We give a simple, direct proof of the easy fact about the Weierstrass Representation, namely, that it always gives a minimal surface. Most presentations include the much harder converse that every simply connected minimal surface is given by the Weierstrass Representation.
The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.
problem Inverse problems on holomorphically separable Kähler manifolds and conformally transversally anisotropic manifolds.
method Application of the Stone-Weierstrass theorem to show uniqueness in inverse problems.
result Generalization and simplification of earlier results in inverse problems.
The paper develops theory for holomorphic null curves in SL2(C).
problem Developing theory for holomorphic null curves in SL2(C).
method Establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for holomorphic null immersions.
result Proves every open Riemann surface admits a proper holomorphic null embedding into SL2(C).
Study identifies specific subvarieties in translation surfaces with quadratic field.
problem Characterizing invariant subvarieties in translation surfaces with quadratic field.
method Analyzing algebraically primitive subvarieties in strata of translation surfaces.
result Only specific subvarieties identified: decagon, Weierstrass curves, etc.
New integrable systems for marginally trapped surfaces in 4D Lorentz-Minkowski space.
problem Constructing new representations for marginally trapped surfaces in L4. method Developed new Weierstrass-type representations to solve a linear PDE.
result Explicit examples of marginally trapped surfaces with non-vanishing mean curvature.
The paper provides Weierstrass representations for minimal space-like surfaces in Minkowski space-time.
problem Minimal space-like surfaces in Minkowski space-time.
method Obtained canonical Weierstrass representations via two holomorphic functions.
result Established geometric correspondence between minimal space-like surfaces and pairs of holomorphic functions.
Short introduction to discrete flat fronts in hyperbolic space with a Weierstrass representation proof.
problem Understanding discrete flat fronts in hyperbolic space.
method Proving a Weierstrass representation for discrete flat fronts.
result Any discrete flat front in the mixed area sense admits a Weierstrass representation.
The paper finds formulas for special surface shapes in 3D space.
problem Creating formulas for constant mean curvature surfaces.
method Weierstrass representations for discrete surfaces in isotropic space.
result Constructs examples of surfaces with discrete parametrizations.
In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, by using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surf…
Researchers describe a Ceresa class for tropical and topological curves, linking algebraic and cohomological perspectives.
problem Explicitly describe the Ceresa class for non-hyperelliptic curves.
method Combining algebraic, tropical, and topological perspectives, defining the Ceresa class for curves and surfaces.
result The Ceresa class is torsion in all settings: tropical curves, topological surfaces, and smooth algebraic curves over C((t)). We find a Weierstrass-like formula for 2D elastic maps.
problem Understanding elastic maps between planar domains.
method Develop a Weierstrass representation for critical points of certain energy functionals.
result Elastic maps admit a Weierstrass representation in terms of holomorphic functions.