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Construction of Triply Periodic Minimal Surfaces
, 1996
"... We discuss the construction of triply period minimal surfaces. This includes concepts for constructing new examples as well as a discussion of numerical computations based on the new concept of discrete minimal surfaces. As a result we present a wealth of old and new examples and suggest directio ..."
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Cited by 10 (2 self)
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We discuss the construction of triply period minimal surfaces. This includes concepts for constructing new examples as well as a discussion of numerical computations based on the new concept of discrete minimal surfaces. As a result we present a wealth of old and new examples and suggest directions for further generalizations.
A FAMILY OF TRIPLY PERIODIC COSTA SURFACES
 PACIFIC JOURNAL OF MATHEMATICS
, 2003
"... We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves. ..."
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Cited by 5 (1 self)
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We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves.
Infinite periodic discrete minimal surfaces without selfintersections
 Balkan J. Geom. Appl
"... Abstract. A triangulated piecewiselinear minimal surface in Euclidean 3space R 3 defined using a variational characterization is critical for area amongst all continuous piecewiselinear variations with compact support that preserve the simplicial structure. We explicitly construct examples of suc ..."
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Abstract. A triangulated piecewiselinear minimal surface in Euclidean 3space R 3 defined using a variational characterization is critical for area amongst all continuous piecewiselinear variations with compact support that preserve the simplicial structure. We explicitly construct examples of such surfaces that are embedded and are periodic in three independent directions of R 3.
Description of cubic liquidcrystalline structures using simple surface foliations
 J. Chem. Soc. Faraday Trans
, 1994
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COPLANAR CONSTANT MEAN CURVATURE SURFACES
, 2007
"... We consider constant mean curvature surfaces with finite topology, properly embedded in threespace in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors [GKS2, GKS1]. Here we extend the arguments to the case of an arbitra ..."
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We consider constant mean curvature surfaces with finite topology, properly embedded in threespace in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors [GKS2, GKS1]. Here we extend the arguments to the case of an arbitrary number of ends, under the assumption that the asymptotic axes of the ends lie in a common plane: we construct and classify the entire family of these genuszero, coplanar constant mean curvature surfaces.
NEW COMPONENTS OF THE MODULI SPACE OF MINIMAL SURFACES IN 4DIMENSIONAL FLAT TORI
, 2004
"... Abstract. In this paper, we consider new components of a key space of a Moduli space of minimal surfaces in flat 4tori and calculate their dimensions. Moreover, we construct an example of minimal surfaces in 4tori and obtain an element of the Moduli. In the process of the construction, we give an ..."
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Abstract. In this paper, we consider new components of a key space of a Moduli space of minimal surfaces in flat 4tori and calculate their dimensions. Moreover, we construct an example of minimal surfaces in 4tori and obtain an element of the Moduli. In the process of the construction, we give an example of minimal surfaces with good property in a 3torus distinct from classical examples. 1.
Triply periodic minimal surfaces bounded by vertical symmetry planes
 Manuscripta Math
"... This material is based upon work for the NSF under Award No.DMS 0139476. Abstract. We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a SchwarzChristoffel formula for periodic polygons in the plane. Our surfaces share ..."
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This material is based upon work for the NSF under Award No.DMS 0139476. Abstract. We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a SchwarzChristoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces. 2000 Mathematics Subject Classification. Primary 53A10; Secondary 49Q05, 53C42. Key words and phrases. Minimal surface, triply periodic, SchwarzChristoffel formula.
Isoperimetric inequalities in crystallography, preprint
, 2003
"... The study of the isoperimetric problem in the presence of crystallographic symmetries is an interesting unsolved question in classical differential geometry: Given a space group G, we want to describe, among surfaces dividing Euclidean 3space into two Ginvariant regions with prescribed volume frac ..."
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The study of the isoperimetric problem in the presence of crystallographic symmetries is an interesting unsolved question in classical differential geometry: Given a space group G, we want to describe, among surfaces dividing Euclidean 3space into two Ginvariant regions with prescribed volume fractions, those which have
The {4, 5} isogonal sponges on the cubic lattice
"... Isogonal polyhedra are those polyhedra having the property of being vertextransitive. By this we mean that every vertex can be mapped to any other vertex via a symmetry of the whole polyhedron; in a sense, every vertex looks exactly like any other. The Platonic solids are examples, but these are bou ..."
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Isogonal polyhedra are those polyhedra having the property of being vertextransitive. By this we mean that every vertex can be mapped to any other vertex via a symmetry of the whole polyhedron; in a sense, every vertex looks exactly like any other. The Platonic solids are examples, but these are bounded polyhedra and our focus here is on infinite polyhedra. When the polygons of an infinite isogonal polyhedron are all planar and regular, the polyhedra are also known as sponges, pseudopolyhedra, or infinite skew polyhedra. These have been studied over the years, but many have been missed by previous researchers. We first introduce a notation for labeling threedimensional isogonal polyhedra and then show how this notation can be combinatorially used to find all of the isogonal polyhedra that can be created given a specific vertex star configuration. As an example, we apply our methods to the {4, 5} vertex star of five squares aligned along the planes of a cubic lattice and prove that there are exactly 15 such unlabeled sponges and 35 labeled ones. Previous efforts had found only 8 of the 15 shapes. 1