1/7/2024 0 Comments Triply twistit mobius space![]() (a) Schwarz’s D-surfaces (b) the surfaces □ 4. By suppressing the catenoidal ends, if we pile up several copies of the fundamental piece, we get the pictures in Figures 1 and 2(b). The fundamental piece resembles the Costa surface with its planar end replaced by either symmetry curves or line segments. Any of those is generated by a fundamental piece, which is a surface with boundary in ℝ 3 with two catenoidal ends. The examples presented herein are inspired in the surfaces □ 2 and □ 2, 4 from. Regarding examples with only vertical symmetries, we believe they have not been found yet. These are equations involving elliptic integrals with interdependent parameters. However, Traizet’s method is not explicit (in the sense explained in ) whereas Fujimori-Weber’s method may turn it hard to analyse the so-called period problems. They are constructed by Karcher’s method, although the purpose alone of few symmetries could be accomplished by modern methods introduced, for instance, by Traizet and Fujimori and Weber. In the present work, we give existence proofs for examples that are probably the first triply periodic minimal surfaces with only horizontal symmetries, of which the translation group is given by an orthogonal lattice. Moreover, when a TT-surface has only horizontal symmetries, its translation group cannot be given by an orthogonal lattice. For edge length 2 √ 3 and height 1, they coincide with the Schwarz P-surface, and hence have further symmetries besides the horizontal ones. The “TT-surfaces” are generated by an annulus, of which the boundary consists of two twisted equilateral triangles. Besides the surfaces shown herein, perhaps there are only the “TT-surfaces” as Karcher named them in (see also ), and a surface from Fischer-Koch, which is, however, presented without rigorous proof. Examples with only horizontal symmetries do not seem to be well known. In fact, this must be true because most of the surfaces in this class have a cubic symmetry group. In the class of triply periodic minimal surfaces almost all known examples have either both or none of such symmetries, after suitable motion in ℝ 3. The doubly periodic examples found by Meeks and Rosenberg in have only vertical symmetries (see also for nice pictures). ![]() For instance, the Costa surface (see ) has only horizontal symmetries. Restricted to symmetries given by reflections in the plane of principal geodesics and by 180°-rotations about straight lines contained in the surface, outside the triply periodic class it is easy to find complete embedded minimal surfaces in ℝ 3 of which these symmetries are either only horizontal or only vertical. However, several symmetry groups are not yet represented by any minimal surface (see for details and comments). Since minimal surfaces may model some natural structures, like crystals and copolymers, an example within a given symmetry group might fit an already existing compound, or even hint at nonexisting ones. With this terminology, he proved that such symmetries induce symmetries in the cone metrics determined by □ ℎ, □ □ ℎ, and □ ℎ / □ from a Weierstraß pair ( □, □ ℎ ) of a minimal surface (see for details).īy classifying the symmetries this way, we sort out the space groups that might admit one, both, or none of them. A vertical symmetry is a reflection at a horizontal plane or a rotation about a vertical line.” “A horizontal symmetry is a reflection at a vertical plane or a rotation about a horizontal line. Weber introduced the following terminology in his first lecture entitled Embedded minimal surfaces of finite topology: Introductionĭuring the Clay Mathematics Institute 2001 Summer School on the Global Theory of Minimal Surfaces, M. We find new examples of embedded triply periodic minimal surfaces for which such symmetries are all of horizontal type. The Schwarz reflection principle states that a minimal surface □ in ℝ 3 is invariant under reflections in the plane of its principal geodesics and also invariant under 180°-rotations about its straight lines.
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