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Classification and Construction of Minimal Translation Surfaces in Euclidean Space

Thomas Hasanis, Rafael López

Results in Mathematics > 2020 > 75 > 1 > 1-22

A translation surface of Euclidean space $${\mathbb {R}}^3$$ R3 is the sum of two regular curves $$\alpha $$ α and $$\beta $$ β , called the generating curves. In this paper we classify the minimal translation surfaces of $${\mathbb {R}}^3$$ R3 and we give a method of construction of explicit examples. Besides the plane and the minimal surfaces of Scherk type, it is proved that up to reparameterizations...

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Translation surfaces with non-zero constant mean curvature in Euclidean space

Thomas Hasanis

Journal of Geometry > 2019 > 110 > 2 > 1-8

We prove that the only surface in 3-dimensional Euclidean space $${\mathbb {R}}^3$$ R3 with constant and non-zero mean curvature H, constructed by the sum of a planar curve and a space curve, is the circular cylinder of radius $$\frac{1}{2|H|}$$ 12|H| .

article

Chamber Structure of Modular Curves $$X_{1}(N)$$ X 1 ( N )

Guillaume Tahar

Arnold Mathematical Journal > 2018 > 4 > 3-4 > 459-481

Modular curves $$X_{1}(N)$$ X 1 ( N ) parametrize elliptic curves with a point of order N. They can be identified with connected components of projectivized strata $$\mathbb {P}\mathcal {H}(a,-a)$$ P H ( a , - a ) of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the topological changes in the flat...

article

Some geometric properties of translating solitons in Euclidean space

Rafael López

Journal of Geometry > 2018 > 109 > 3 > 1-15

A translating soliton is a surface in Euclidean space $$\mathbb {R}^3$$ R3 that is minimal for a log-linear density $$\phi (x,y,z)=\alpha x+\beta y+\gamma y$$ ϕ(x,y,z)=αx+βy+γy , where $$\alpha ,\beta ,\gamma $$ α,β,γ are real numbers not all zero. We characterize all translating solitons that are foliated by circles and the ones that are graphs of type $$z=f(x)+g(y)$$ z=f(x)+g(y) .

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Counting saddle connections in flat surfaces with poles of higher order

Guillaume Tahar

Geometriae Dedicata > 2018 > 196 > 1 > 145-186

Flat surfaces that correspond to k-differentials on compact Riemann surfaces are of finite area provided there is no pole of order k or higher. We denote by flat surfaces with poles of higher order those surfaces with flat structures defined by a k-differential with at least one pole of order at least k. Flat surfaces with poles of higher order have different geometrical and dynamical properties than...

article

Minimal Translation Surfaces in Euclidean Space

Rafael López, Óscar Perdomo

The Journal of Geometric Analysis > 2017 > 27 > 4 > 2926-2937

A translation surface in Euclidean space is a surface that is the sum of two regular curves $$\alpha $$ α and $$\beta $$ β . In this paper we characterize all minimal translation surfaces. In the case that $$\alpha $$ α and $$\beta $$ β are non-planar curves, we prove that the curvature $$\kappa $$ κ and the torsion $$\tau $$ τ of both curves must...

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Weighted minimal translation surfaces in the Galilean space with density

Dae Won Yoon

Open Mathematics > 2017 > 15 > 1 > 459-466

Translation surfaces in the Galilean 3-space G3 have two types according to the isotropic and non-isotropic plane curves. In this paper, we study a translation surface in G3 with a log-linear density and classify such a surface with vanishing weighted mean curvature.

article

Residual generic ergodicity of periodic group extensions over translation surfaces

David Ralston, Serge Troubetzkoy

Geometriae Dedicata > 2017 > 187 > 1 > 219-239

Continuing the work in Ralston and Troubetzkoy (J Mod Dyn 6:477–497, 2012), we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group extension produced using a technique referred to as cuts.

article

Lattice surfaces and smallest triangles

Chenxi Wu

Geometriae Dedicata > 2017 > 187 > 1 > 107-121

We calculate the area of the smallest triangle and the area of the smallest virtual triangle for many known lattice surfaces. We show that our list of the lattice surfaces for which the area of the smallest virtual triangle greater than $$1\over 20$$ 1 20 is complete. In particular, this means that there are no new lattice surfaces for which the area of the smallest virtual triangle is...

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Minimal translation hypersurfaces in E4

Marilena Moruz, Marian Ioan Munteanu

Journal of Mathematical Analysis and Applications > 2016 > 439 > 2 > 798-812

In this paper we consider hypersurfaces in the Euclidean space E4 defined as the sum of a curve and a surface whose mean curvature vanishes. We call them minimal translation hypersurfaces in E4 and give a classification of these hypersurfaces.

article

A generalization of translation surfaces with constant curvature in the isotropic space

Muhittin Evren Aydin

Journal of Geometry > 2016 > 107 > 3 > 603-615

In this paper we study the translation surfaces generated by a space curve and a planar curve in the isotropic 3-space $${\mathbb{I}^{3}}$$ I 3 . We completely classify such surfaces in $${\mathbb{I}^{3}}$$ I 3 with constant curvature. Several examples are also given by figures.

article

Periodic billiard orbits of self-similar Sierpiński carpets

Joe P. Chen, Robert G. Niemeyer

Journal of Mathematical Analysis and Applications > 2014 > 416 > 2 > 969-994

We identify a collection of periodic billiard orbits in a self-similar Sierpiński carpet billiard table Ω(Sa). Based on our refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in Sa, we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpiński carpet billiard tables Ω(Sa,n). The trivial limit of this sequence...

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Translation surfaces of constant curvatures in a simply isotropic space

Željka Milin Šipuš

Periodica Mathematica Hungarica > 2014 > 68 > 2 > 160-175

In this paper we describe, up to a congruence, translation surfaces in a simply isotropic space having constant isotropic Gaussian or mean curvature. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature $$K\ne 0$$ K ≠ 0 and translation surfaces with constant mean curvature $$H\ne 0$$ H ≠ 0 that are not cylindrical...

article

Affine cylinders

V. N. Kokarev

Mathematical Notes > 2014 > 96 > 5-6 > 686-689

An affine cylinder is defined as a surface for which all affine normals are parallel to the same plane. It is proved that a complete strictly convex affine cylinder is a translation surface whose line of translation is a parabola.

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Hyperelliptic translation surfaces and folded tori

Chris Johnson, Martin Schmoll

Topology and its Applications > 2014 > 161 > Complete > 73-94

Following constructions of McMullen [11] and Vasilyev [15] we list strict quadratic differentials on (marked) tori defined by twists of one-forms on hyperelliptic surfaces. We count the total number of twists in each genus and show that the quadratic differentials on (marked) tori are strict. Taking universal covers we obtain a variety of (non-arithmetic) lattice Panov planes [14] from genus two lattice...

article

On the Teichmüller geodesic generated by the L-shaped translation surface tiled by three squares

Olivier Rodriguez

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales.... > 2013 > 107 > 2 > 391-417

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices. We show that the only Riemann surface in that family admitting a non-hyperelliptic...

article

The complete family of Arnoux–Yoccoz surfaces

Joshua P. Bowman

Geometriae Dedicata > 2013 > 164 > 1 > 113-130

The family of translation surfaces (X _g, ω _g) constructed by Arnoux and Yoccoz from self-similar interval exchange maps encompasses one example from each genus g greater than or equal to 3. We triangulate these surfaces and deduce general properties they share. The surfaces (X _g, ω _g) converge to a surface (X _∞, ω _∞) of infinite genus and finite area. We study the exchange on infinitely many...

article

Ellipses in translation surfaces

S. Allen Broughton, Chris Judge

Geometriae Dedicata > 2012 > 157 > 1 > 111-151

We study the topology and geometry of the moduli space of immersions of ellipses into a translation surface. The frontier of this space is naturally stratified by the number of ‘cone points’ that an ellipse meets. The stratum consisting of ellipses that meet three cone points is naturally a two dimensional (non-manifold) polygonal cell complex. We show that the topology of this cell-complex together...

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Minimal translation surfaces in the Heisenberg group Nil₃

Jun-ichi Inoguchi, Rafael López, Marian-Ioan Munteanu

Geometriae Dedicata > 2012 > 161 > 1 > 221-231

A translation surface in the Heisenberg group Nil₃ is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group Nil₃.

article

Surfaces with constant mean curvature in Sol geometry

Rafael López, Marian Ioan Munteanu

Differential Geometry and its Applications > 2011 > 29 > Supplement 1 > S238-S245

Among the eight geometries of Thurston, Sol ₃ is the space with the smallest number of isometries, for example, there are no rotations. In this work, we classify all surfaces with constant mean curvature that either are invariant by a 1-parameter group of isometries or are the product of two planar curves.

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Keywords

TRANSLATION SURFACE (28)
MINIMAL SURFACE (6)
MEAN CURVATURE (3)
VEECH GROUP (3)
CENTROAFFINE DIFFERENTIAL GEOMETRY (2)
FLAT STRUCTURE (2)
FLAT SURFACE (2)
GAUSSIAN CURVATURE (2)
JACOBI EQUATION (2)
QUADRATIC DIFFERENTIAL (2)
SECOND GAUSSIAN CURVATURE (2)
WEINGARTEN SURFACE (2)
(EVENTUALLY) CONSTANT SEQUENCES OF COMPATIBLE ORBITS (1)
ABELIAN DIFFERENTIAL (1)
AFFINE CYLINDER (1)
AFFINE HOMEOMORPHISM (1)
AFFINE NORMAL (1)
ALGEBRAIC CURVE (1)
BILLIARD FLOW (1)
BRANCHED COVERS (1)
CIRCULAR CYLINDER (1)
COMPLETE STRICTLY CONVEX AFFINE CYLINDER (1)
CONSTANT MEAN CURVATURE (1)
COVERING (1)
CURVES (1)
DARBOUX SURFACE (1)
EHRENFEST WIND-TREE MODEL (1)
ERGODICITY (1)
FOOTPRINTS (1)
FRACTAL (1)
FRACTAL BILLIARD (1)
GALILEAN SPACE (1)
GAUSS CURVATURE (1)
GEODESIC FLOW (1)
HEISENBERG GROUP (1)
HELICOIDAL SURFACE (1)
HESSIAN MATRIX (1)
HIGHER ORDER DIFFERENTIAL (1)
HYPERBOLIC SPACE (1)
HYPERELLIPTIC SURFACES (1)
INTERVAL EXCHANGE TRANSFORMATION (1)
INVARIANT SURFACE (1)
ISOTROPIC MEAN CURVATURE (1)
ISOTROPIC MINIMAL (1)
ISOTROPIC SPACE (1)
ITERATED FUNCTION SYSTEM AND ATTRACTOR (1)
LATTICE SURFACE (1)
LAW OF REFLECTION (1)
LOG-LINEAR DENSITY (1)
LORENTZ GAS (1)
MODULAR CURVES (1)
MODULI SPACE (1)
PERIOD MATRIX (1)
PICK INVARIANT (1)
POLYGONAL BILLIARD (1)
PREFRACTAL APPROXIMATIONS (1)
PREFRACTAL RATIONAL BILLIARD APPROXIMATIONS (1)
QUADRATIC FORM (1)
RATIONAL (POLYGONAL) BILLIARD (1)
RECURRENCE (1)
RELATIVE CURVATURE (1)
RIEMANN SURFACE (1)
RIEMANN SURFACES (1)
SADDLE CONNECTION (1)
SCHERK SURFACE (1)
SELF-SIMILAR SET (1)
SELF-SIMILAR SIERPIŃSKI CARPET BILLIARD (1)
SEQUENCE OF COMPATIBLE ORBITS (1)
SIMPLY ISOTROPIC SPACE (1)
SKEW PRODUCT (1)
SOL3 (1)
SPACE CURVE (1)
SQUARE TILED SURFACE (1)
TEICHMUELLER DISC (1)
TEICHMÜLLER CURVE (1)
TEICHMÜLLER GEODESIC (1)
THETA CHARACTERISTIC (1)
THURSTON–VEECH CONSTRUCTION (1)
TRANSLATING SOLITON (1)
TRIANGULATION (1)
UMBILICAL SURFACE (1)
UNFOLDING PROCESS (1)
WALLS-AND-CHAMBERS STRUCTURE (1)
WEIGHTED MANIFOLD (1)
WEIGHTED MEAN CURVATURE (1)
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Search results

Classification and Construction of Minimal Translation Surfaces in Euclidean Space

Translation surfaces with non-zero constant mean curvature in Euclidean space

Chamber Structure of Modular Curves $$X_{1}(N)$$ X 1 ( N )

Some geometric properties of translating solitons in Euclidean space

Counting saddle connections in flat surfaces with poles of higher order

Minimal Translation Surfaces in Euclidean Space

Weighted minimal translation surfaces in the Galilean space with density

Residual generic ergodicity of periodic group extensions over translation surfaces

Lattice surfaces and smallest triangles

Minimal translation hypersurfaces in E4

A generalization of translation surfaces with constant curvature in the isotropic space

Periodic billiard orbits of self-similar Sierpiński carpets

Translation surfaces of constant curvatures in a simply isotropic space

Affine cylinders

Hyperelliptic translation surfaces and folded tori

On the Teichmüller geodesic generated by the L-shaped translation surface tiled by three squares

The complete family of Arnoux–Yoccoz surfaces

Ellipses in translation surfaces

Minimal translation surfaces in the Heisenberg group Nil₃

Surfaces with constant mean curvature in Sol geometry

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