The paper classifies affine minimal translation surfaces and finds their properties.
problem Classifying and understanding affine minimal translation surfaces.
method Using Weierstrass-Enneper formula and hodographic coordinate system.
result Classification and properties of affine minimal translation surfaces.
The paper studies a new class of affine maximal surfaces with singularities.
problem Understanding the properties of affine maximal surfaces with singularities.
method Defining a new subclass of affine maximal surfaces and applying Euclidean minimal surface theory.
result Affine maxfaces satisfy an Osserman-type inequality and do not contain non-trivial improper affine fronts.
We investigate the geometric properties of hyperbolic affine flat, affine minimal surfaces in the equiaffine space A3. We use Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of such surfaces and…
Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that they are critical points of an affine area functional defined on the space of qu…
The paper studies singularities in discrete indefinite affine minimal surfaces.
problem Characterizing singularities in discrete indefinite affine minimal surfaces.
method Discretizing smooth curves and applying discrete Lelieuvre's formulas to study the resulting surfaces.
result The definition of singular edges and vertices in discrete asymptotic nets mirrors properties of smooth surfaces.
In this paper we study the general affine differential geometry of surfaces in affine space A3. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to aff…
The paper extends Bäcklund theorem in affine differential geometry of surfaces.
problem Understanding local symmetries and proportional affine fundamental forms in surfaces.
method Formulating and proving complementary affine Bäcklund theorem.
result Conditions for local symmetry and proportional affine fundamental forms.
Minimal surfaces in Heisenberg group have null curves and lines.
problem Characterizing timelike minimal surfaces in the Heisenberg group.
method Characterization through null curves and lines with prescribed curvatures.
result Minimal surfaces are defined by the multiplication of null curves and affine null lines.
Lie minimal surfaces are characterized by differential equations of principal curvatures.
problem Characterizing Lie minimal surfaces in Riemannian space forms.
method Using Euler-Lagrange equations and differential equations of principal curvatures.
result Rotational surfaces are found for certain relationships between principal curvatures.
Paper classifies timelike Thomsen surfaces and reveals their geometric properties.
problem Classifying timelike Thomsen surfaces and understanding their geometric properties.
method Use of Lorentz conformal and null coordinates, representation theorems, and geometric invariants.
result Revealed the relationship between timelike Thomsen surfaces and timelike minimal surfaces with planar curvature lines.
Study classifies zero mean curvature surfaces with planar curvature lines.
problem Characterizing surfaces with specific curvature properties.
method Complete classification and investigation of their relationship to Thomsen-type surfaces.
result Zero mean curvature surfaces with planar curvature lines belong to a 1-parameter family.
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 the…
Study proves no nontrivial minimal surfaces in half-space with specific boundary conditions.
problem Proving the nonexistence of minimal surfaces in half-space with certain boundary conditions.
method Analyzes minimal surface equations in half-space with specific boundary conditions.
result Establishes Liouville type theorems for minimal surfaces in half-space.
We discuss the local differential geometry of convex affine spheres in $\re^3$ and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick f…
Algorithm finds periodic points on Veech surfaces.
problem Finding periodic points on non-square-tiled Veech surfaces.
method Developed an algorithm to compute periodic points.
result Proved that in low discriminant, non-square-tiled Veech surfaces have no periodic points, except for fixed points of the Prym involution.
In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.
Simple proof shows Bernstein property fails for a minimal surface equation.
problem Whether a specific minimal surface equation has the Bernstein property.
method Simple argument to show the equation does not have the Bernstein property.
result The minimal surface equation does not have the Bernstein property.
Study shows periodic points of Prym eigenforms in specific genera.
problem Understanding periodic points of Prym eigenforms in translation surfaces.
method Geometric proof using Prym involution and affine automorphism group.
result Fixed points of Prym involution are periodic points of Prym eigenforms.
The paper studies timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
problem Characterizing timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
method Defining natural Gauss maps and proving a Ruh-Vilms type theorem.
result Timelike minimal Lagrangian surfaces correspond to the fifth real form of the complex affine Kac-Moody algebra.
We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean n-space (n=3 or 4), improper affine spheres in the affine 3-space and flat surfaces in hyperbolic 3-space. In parti…
The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is closely related to normal curvature, and from this we re-calculate the Minkowski Gau…
The paper proves conditions for minimal surfaces to be holomorphic and stable.
problem Conditions for stable minimal surfaces to be holomorphic.
method Developed a method of constructing variations to prove the equivalence.
result Holomorphicity and stability conditions for minimal surfaces.
Survey on real forms of a complex equation and their connection to surface theory.
problem Describing real forms of the complex A2(2)-Toda equation and their geometric implications. method Analyzing the integrability of Maurer-Cartan forms for different real forms of loop groups.
result Each real form of A2(2) corresponds to a specific surface class with integrable frames. The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space and wea…
We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat…
Characterizes stable minimal capillary surfaces with specific angles.
problem Understanding stable minimal capillary surfaces with near 0 or π angles. method Curvature estimates for sequences of weakly stable minimal capillary surfaces.
result Characterization of tangential limits of stable minimal capillary surfaces.
It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Frechet manifold (if it is not the empty set). The moduli space of all connected compact orientable metric Lagrangian embedded surfaces…
Minimal surfaces with isothermal parameters admitting Bézier representation were studied by Cosin and Monterde. They showed that, up to an affine transformation, the Enneper surface is the only bi-cubic isothermal minimal surface. Here we study bi-quartic isothermal minimal surfaces and establish the general form of th…
Entire area-minimizing surfaces of density 2 are planar or quadratic
problem Classifying entire area-minimizing surfaces
method Using density and algebraic properties
result All such surfaces are planar or algebraic
Almost Zoll affine surface found on cylinder.
problem Finding surfaces with special geodesic properties.
method Exhibited an affine structure on a cylinder.
result Affine structure on cylinder is almost Zoll.
Unified approach to totally ramified values in various surface theories.
problem Totally ramified values in value distribution theory, normal family theory, and Gauss maps of surfaces.
method Bloch--Ros principle applied to various surface theories.
result Unified approach to phenomena concerning totally ramified values.
Defines CAMC discrete nets and their properties.
problem Understanding CAMC discrete nets and their properties.
method Defining CAMC discrete nets and proving properties.
result Properties of CAMC discrete nets are equivalent to properties of compatible interpolating quadrics.
Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for Lφ affine surface areas are established.
Study geodesic and affine Killing completeness in homogeneous affine surfaces.
problem Geodesic and affine Killing completeness in homogeneous affine surfaces.
method Examined using the solution space of the quasi-Einstein equation.
result Characterized geodesic and affine Killing completeness in homogeneous affine surfaces.
Proofs for decomposing branched affine surfaces into triangles and cylinders.
problem Decomposing branched affine surfaces into simpler geometric shapes.
method Proof of Veech's theorem and introduction of invariant α.
result Any pair of decompositions can be connected by flips.
Overview of affine surface area and its history.
problem None explicitly stated; focuses on overview.
method None explicitly stated; focuses on overview.
result None explicitly stated; focuses on overview.
Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X(S, PSL(2,C)) and the affine cotangent symplectic structure on the space of complex projective structures CP(S) given by t…
Algorithm constructs surfaces with specific Veech groups in lattice strata.
problem Finding all translation surfaces with a given lattice Veech group.
method Developed an algorithm and provided a new proof of finiteness.
result Algorithm constructs all translation surfaces with a given lattice Veech group.
Identifies holonomy of affine surfaces via meromorphic connections.
problem Holonomy of complex affine surfaces.
method Moduli space of complex affine surfaces and meromorphic connections.
result Holonomy map is a submersion for non-finite-area translation surfaces.
Study affine curvature lines on surfaces in 3D space.
problem Understanding the behavior of affine curvature lines on surfaces.
method Analyzing binary differential equations and topological models.
result Obtained topological models and generic behavior of affine curvature lines.
Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late 19th century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only recently been given. In this paper, we use an alternative method not only to re…
Introduces new weighted floating functions and affine surface areas.
problem Developing new mathematical concepts for convex bodies.
method Introducing weighted floating functions and weighted functional affine surface areas.
result New relations to traditional and classical affine surface areas.
Paper generalizes Bloch-Ros principle to various surface classes.
problem Understanding the relationship between normal family theory, value distribution theory, and surface theory.
method Formulation and generalization of Bloch-Ros principle to different surface classes.
result Effective criterion for determining Gaussian curvature estimates for various surface classes.
Constructs moduli spaces for complex affine and dilation surfaces.
problem Classifying and understanding moduli spaces of complex surfaces.
method Using Veech's ideas, constructs holomorphic affine bundles and covering spaces.
result Moduli spaces of dilation surfaces are orbifold K(G,1) where G is the framed mapping class group.
The study examines stable minimal surfaces in higher dimensions and provides bounds on their properties.
problem Properties of stable minimal surfaces in higher codimension.
method Structural analysis of holomorphic vector bundles and geometric inequalities.
result Explicit bounds on the systole for stable minimal tori and surfaces.
The paper classifies compact affine quaternionic curves and surfaces.
problem Classifying compact affine quaternionic curves and surfaces.
method Affine quaternionic manifolds, Kodaira Theorem, fundamental groups, Lie Groups.
result Only quaternionic tori and primary Hopf surface S^3 x S^1 are compact affine quaternionic curves.
We prove that, given a compact Riemann surface Σ and disjoint finite sets ∅=E⊂Σ and Λ⊂Σ, every map Λ→R3 extends to a complete conformal minimal immersion Σ∖E→R3 with finite total curvature. This result opens the door to study optimal hitting problem…
New maximal surfaces solve Bernstein problems.
problem Bernstein problems in centroaffine geometry.
method Calabi affine maximal surfaces and orthonormal frame fields.
result Complete centroaffine extremal hypersurfaces solve all Bernstein problems.