We construct triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space, with the same topology as the triply periodic minimal surfaces in the Euclidean 3-space, called Schwarz rPD surfaces.
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48 results for “triply-periodic minimal surfaces”
Two new triply periodic minimal surfaces of genus 4 discovered.
problem Finding new triply periodic minimal surfaces of genus 4.
method Combination of asymptotic analysis and geometric methods to solve the period problem.
result Two new 1-parameter families of embedded triply periodic minimal surfaces of genus 4.
We use bifurcation theory to determine the existence of infinitely many new examples of triply periodic minimal surfaces in . These new examples form branches issuing from the H-family, the rPD-family, the tP-family, and the tD-family, that converge to some degenerate embedding of the families. As to nonde…
We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces.
New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.
problem Limiting the outcomes of gluing Scherk surfaces into minimal surfaces.
method Constructing minimal surfaces by stacking and gluing doubly periodic Scherk surfaces.
result Except for special cases, gluing more Scherk surfaces results in known minimal surfaces.
We find the first examples of triply periodic minimal surfaces of which the intrinsic symmetries are all of horizontal type.
Researchers create new triply periodic minimal surfaces by gluing saddle towers.
problem Creating triply periodic minimal surfaces without symmetry constraints.
method Gluing Karcher-Scherk saddle towers with phase differences and balancing under vertical interaction.
result Expands known triply periodic minimal surfaces into new 5-parameter families.
We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain of 3-space with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices . The network…
Given a tiling of the plane by straight edge polygons, which is invariant by two independent translations, we construct a family of embedded triply periodic minimal surfaces which desingularizes . For this purpose, inspired by the work of Martin Traizet, we open the nodes of s…
Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfacesmath.GT
New method for counting distinct tilings with symmetrical surfaces.
problem Counting distinct tilings with symmetrical surfaces.
method Deriving representations of mapping class groups and describing tilings as decorations on orbifolds.
result Explicit enumeration of isotopically distinct tilings.
Existence proof for two families of triply periodic minimal surfaces.
problem Proving the existence of two families of minimal surfaces.
method Extending a technique to non-rectangular branched tori.
result Existence of the tG and rGL families of triply periodic minimal surfaces.
We get a continuous one-parameter new family of embedded minimal surfaces, of which the period problems are two-dimensional. Moreover, one proves that it has Scherk second surface and Hoffman-Wohlgemuth example as limit-members.
The study constructs periodic surfaces using graph theory and applies cyclically branched coverings to identify their conformal type.
problem Constructing and identifying the conformal type of periodic surfaces with a given geometric structure.
method Graph theory and cyclically branched coverings.
result Explicit cone metrics on compact Riemann surfaces can be realized as the quotient of triply periodic polyhedral surfaces.
We describe a new family of triply-periodic minimal surfaces with hexagonal symmetry, related to the quartz (qtz) and its dual (the qzd net). We provide a solution to the period problem and provide a parametrisation of these surfaces, that are not in the regular class, by the Weierstrass-Enneper formalism. We identifie…
In the previous work, the first author established an algorithm to compute the Morse index and the nullity of an -periodic minimal surface in . In fact, the Morse index can be translated into the number of negative eigenvalues of a real symmetric matrix and the nullity can be translated into the number…
Discuss Alan Schoen's I-WP minimal surface with geometric realizations.
problem Characterize and visualize Alan Schoen's I-WP minimal surface.
method Examine various geometric realizations of the I-WP surface and analyze the associate family.
result The associate family of the I-WP surface contains six congruent surfaces at specific angles.
Embedded triply periodic zero mean curvature surfaces of mixed type in Lorentz-Minkowski 3-spacemath.DG
We construct embedded triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space with the same topology as the Schwarz D surface in the Euclidean 3-space.
Paper constructs and proves existence of chiral triply-periodic minimal surfaces.
problem Existence of chiral triply-periodic minimal surfaces of genus 4.
method Construction based on dual qtz--qzd nets, existence proof using Weierstrass method on branched torus.
result Family of chiral triply-periodic minimal surfaces of genus 4 constructed and proved to exist.
We prove the existence of a new 2-parameter family o of embedded triply periodic minimal surfaces of genus 3. The new surfaces share many properties with classical orthorhombic deformations of Schwarz' D surface, but also exotic in many ways. In particular, they do not belong to Meeks' five-dimensional family. Never…
Given a closed flat 3-torus , for each and each non-negative integer , we obtain area estimates for closed surfaces with genus and constant mean curvature embedded in . This result contrasts with the theorem of Traizet [33], who proved that every flat 3-torus admits for every positive integer …
Minimal Twin Surfacesmath.DG
We report some minimal surfaces that can be seen as copies of a triply periodic minimal surface (TPMS) related by reflections in parallel mirrors. We call them minimal twin surfaces for the resemblance with twin crystal. Brakke's Surface Evolver is employed to construct twinnings of various classical TPMS, including Sc…
This study investigates porosity and topological properties of TPMS using machine learning.
problem Understanding the relationships between porosity and topological properties of TPMS.
method Application of machine learning techniques to analyze porosity and shape factor of TPMS.
result Conjectures suggesting polynomial relationships between porosity and shape factor of TPMS.
Schwarz H surfaces can be continuously deformed into Meeks surfaces.
problem Classifying and understanding the relationship between Schwarz H surfaces and Meeks surfaces.
method Constructing a 2-parameter family of TPMS of genus three that contains both H and Meeks surfaces.
result Schwarz H surfaces can be continuously deformed into Meeks surfaces.
In this paper, we will construct an example of a closed Riemann surface that can be realized as a quotient of a triply periodic polyhedral surface where the Weierstrass points of coincide with the vertices of First we construct by attaching Platonic solids in a periodic manner a…
Transforming cylindrical packings into bicontinuous surfaces.
problem Understanding the early development of bicontinuous structures in plant plastids.
method Geometric modeling and computational simulations of cylinder packings.
result Specific cylinder packings with cubic symmetry transform into TPMS.
Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.
problem Creating space-filling shapes without sharp corners.
method Edge bending algorithm to deform polyhedral tilings into soft tilings.
result Soft tilings derived from minimal surfaces can be continuously transformed into one another.
Weaved helices form mechanically stable 3D structures.
problem Creating stable 3D structures from helical elements.
method Exploiting screw symmetry and invariant cylindrical rod packing to form triply periodic arrangements.
result Demonstrated nineteen triply periodic arrangements of interwoven helices.
The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In th…
Explains connections between monopoles and modules on elliptic curves.
problem Understanding relationships between different mathematical objects.
method Explains equivalences between monopoles and polystable bundles and modules.
result Monopoles and polystable difference modules on elliptic curves are equivalent.
Biplanar Foldingsmath.DG
New framework for folding polyhedra into two planes.
problem Creating polyhedra that can be folded into two planes.
method General framework for bifoldable polyhedra construction.
result Examples of infinite triply periodic and fractal bifoldable polyhedra.
Paper defines linking numbers for periodic tangles.
problem Defining invariant linking numbers for periodic tangles.
method Defined linking numbers using motifs and showed invariance.
result Linking numbers are well-defined and invariant.
New diagrams classify triply periodic entanglements.
problem Classifying triply periodic entanglements.
method Introducing diagrams for links in 3-torus and defining additional moves.
result New diagrams allow classification of non-isotopic links with same preimage.
Study on minimal surfaces in a 3D space with 2m-norm.
problem Characterizing minimal surfaces in a specific geometric space.
method Examining translation, homothetical, and separable minimal surfaces.
result New insights into minimal surfaces in a 3D space with 2m-norm.
This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
problem Compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
method Proves compactness through bubble tree limit analysis.
result Compactness of conformal Chern-minimal surfaces is established with bounded area.
Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
Minimal surfaces in 3-sphere created by reflections from polygons, with new examples based on pentagons.
problem Constructing minimal surfaces in 3-sphere using reflections.
method Minimal -gon solves free boundary problem; curvature lines combinatorics investigated.
result New examples of minimal reflection surfaces based on pentagons.
Study minimal surfaces in Kropina 3D space, finding only planes as minimal translation surfaces.
problem Characterizing minimal surfaces in Kropina 3D space.
method Solving partial differential equations to characterize minimal surfaces.
result Only planes are minimal translation surfaces in Kropina 3D space.
Paper proves stable minimal surfaces in 3D are flat.
problem Understanding stable minimal surfaces in 3D.
method Analyzes quadratic area growth and stability conditions.
result Stable minimal Plateau surfaces in 3D are flat.
Ruh-Vilms Theorems For Minimal Surfaces Without Complex Points and Minimal Lagrangian Surfaces in math.DG
The paper characterizes minimal surfaces and Lagrangian surfaces in complex projective space.
problem Characterizing minimal surfaces and Lagrangian surfaces in complex projective space.
method Using Ruh-Vilms type theorems.
result Characterizes minimal surfaces and Lagrangian surfaces in complex projective space.
New inequality helps map stability in minimal surfaces.
problem Stability of minimal surfaces in .
method Developing new inequalities and perspectives on minimal surfaces.
result Reproves instability of classical minimal surfaces like Enneper.
Minimal surfaces help compare geometric shapes.
problem Comparing different geometric shapes.
method Applications of minimal surfaces.
result New insights into geometric comparisons.
New minimal surfaces found using Toda lattice and integrable systems.
problem Constructing new minimal surfaces with specific genus.
method Using Toda lattice and integrable systems techniques.
result New singly periodic minimal surfaces with genus .
This paper connects Laguerre minimal surfaces to Weierstrass representations.
problem Understanding the relationship between Laguerre minimal surfaces and Weierstrass representations.
method Defining spherical mean curvature and providing Weierstrass-type representations for two classes of surfaces.
result Laguerre minimal surfaces are related to -surfaces, providing a new Weierstrass-type representation.
We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for st…
Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.
problem Existence of minimal surfaces in non-compact 3-manifolds.
method Analyzes specific cases of hyperbolic 3-manifolds with bounded geometry and rank-1 cusps.
result Proves existence of minimal surfaces in various hyperbolic 3-manifolds.
Unstable minimal surfaces in n-space link to hyperbolic products.
problem Characterizing unstable minimal surfaces in and their product counterparts.
method Lifting to -trees, deforming to hyperbolic products, and proving instability equivalence.
result Unstable minimal surfaces in imply unstable surfaces in product hyperbolic spaces.
Maximal surfaces in correspond to timelike minimal surfaces.
problem Establishing a correspondence between maximal and timelike minimal surfaces in .
method Linear transformation between maximal surfaces and timelike minimal surfaces, preserving singularities and Gauss map.
result One-one correspondence and preservation of properties between maximal and timelike minimal surfaces.
The study examines minimal surfaces in Riemannian products of surfaces.
problem Exploring geometric and topological restrictions on minimal surfaces in Riemannian products of surfaces.
method Analyzes totally geodesic surfaces and minimal 2-spheres, 2-tori, and 2-spheres in Riemannian products of surfaces with constant curvature.
result Generically, a totally geodesic surface in a Riemannian product is either a slice or a product of geodesics. Minimal 2-spheres and 2-tori have specific properties under certain curvature conditions.