In this paper, we consider minimal graphs in the three-dimensional Riemannian manifold M×R. We mainly estimate the Gaussian curvature of such surfaces. We consider the minimal disks and minimal ...graphs bounded by two Jordan curves in parallel planes. The key to the proofs is the Weierstrass representation of those surfaces via ℘-harmonic mappings. We also prove some Schwarz lemma type results and some Heinz type results for harmonic mappings between geodesic disks in Riemannian surfaces.
We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space SL˜2(R) with isometry group of dimension 4, in terms of their ...asymptotic boundary. Also, we show that a properly immersed minimal surface in SL˜2(R) contained between two bounded entire minimal graphs separated by vertical distance less than 1+4τ2π has multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a family of complete embedded minimal surfaces which are non-proper in SL˜2(R).
Nature‐inspired materials based on triply periodic minimal surfaces (TPMS) are very attractive in many engineering disciplines because of their topology‐driven properties. However, their adoption ...across different research and engineering fields is limited by the complexity of their design process. In this work, we present MSLattice, a software that allows users to design uniform, and functionally grade lattices and surfaces based on TPMS using two approaches, namely, the sheet networks and solid networks. The software allows users to control the type of TPMS topology, relative density, cell size, relative density grading, cell size grading, and hybridization between lattices. These features make MSLattice a complete design platform for users in different engineering disciplines, especially in applications that employ additive manufacturing (3D printing) and computational modeling. We demonstrate the capability of the software using several examples.
We prove Schwarz-type lemma results for Weierstrass parameterization of the minimal disk in the Riemannian manifold M×R, where M is a Riemannian surface of non-positive Gaussian curvature. The ...estimate is sharp, and the equality is attained if and only if the ϱ-harmonic mapping that produces the parameterization is conformal and the metric is a Euclidean metric. If the area of the minimal surface is equal to the area of the disk, then the parametrization is a contraction w.r.t. induced metric and hyperbolic metric respectively.
Three-dimensional lattices have applications across a range of fields including structural lightweighting, impact absorption and biomedicine. In this work, lattices based on triply periodic minimal ...surfaces were produced by polymer additive manufacturing and examined with a combination of experimental and computational methods. This investigation elucidates their deformation mechanisms and provides numerical parameters crucial in establishing relationships between their geometries and mechanical performance. Three types of lattice were examined, with one, known as the primitive lattice, being found to have a relative elastic modulus over twice as large as those of the other two. The deformation process of the primitive lattice was also considerably different from those of the other two, exhibiting strut stretching and buckling, while the gyroid and diamond lattices deformed in a bending dominated manner. Finite element predictions of the stress distributions in the lattices under compressive loading agreed with experimental observations. These results can be used to create better informed lattice designs for a range of mechanical and biomedical applications.
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•Manufactured and tested lattice structures based on triply periodic minimal surfaces.•Lattices with equivalent masses deform differently depending on their cell geometry.•High stiffness seen for the structure which showed buckling and low failure strain.•Determined Gibson-Ashby factors enabling the design of optimised latticed components.
Triply periodic minimal surface structures and related geometries are widely identified in many natural systems, such as biological membranes and biophotonic structures in butterfly‐wing scales. ...Inspired by their marvelous and highly symmetrical structures and optimized physical properties, these structures have sparked immense interest for creating novel materials by extracting the design from nature. Significant progress has been made to understand these biological structures and fabricate artificial materials by top‐down and bottom‐up approaches for numerous applications in chemistry and materials science. Herein, research achievements, including theoretical and experimental discoveries, in both biological systems and the artificial synthesis of materials with triply periodic minimal surface structures and related materials are summarized. Recent developments in self‐assembled lyotropic liquid crystal phases, block copolymer systems, and their inorganic replicas are discussed in detail.
Triply periodic minimal surfaces and related materials have attracted great attention due to their highly symmetrical structures and optimized properties. The formation, properties, and applications of related materials in natural and artificial systems are summarized. Lyotropic liquid‐crystal phases, block copolymer systems, and their inorganic replicas are emphasized and discussed in detail.
This article offers a comprehensive overview of the results obtained through numerical methods in solving the minimal surface equation, along with exploring the applications of minimal surfaces in ...science, technology, and architecture. The content is enriched with practical examples highlighting the diverse applications of minimal surfaces.
Abstract
Inspired by natural porous architectures, numerous attempts have been made to generate porous structures. Owing to the smooth surfaces, highly interconnected porous architectures, and ...mathematical controllable geometry features, triply periodic minimal surface (TPMS) is emerging as an outstanding solution to constructing porous structures in recent years. However, many advantages of TPMS are not fully utilized in current research. Critical problems of the process from design, manufacturing to applications need further systematic and integrated discussions. In this work, a comprehensive overview of TPMS porous structures is provided. In order to generate the digital models of TPMS, the geometry design algorithms and performance control strategies are introduced according to diverse requirements. Based on that, precise additive manufacturing methods are summarized for fabricating physical TPMS products. Furthermore, actual multidisciplinary applications are presented to clarify the advantages and further potential of TPMS porous structures. Eventually, the existing problems and further research outlooks are discussed.
In this paper, we study the Gieseker moduli space M1,14,3 of minimal surfaces with pg=q=1,K2=4 and genus 3 Albanese fibration. Under the assumption that direct image of the canonical sheaf under the ...Albanese map is decomposable, we find two irreducible components of M1,14,3, one of dimension 5 and the other of dimension 4.
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