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In this paper, we study the hyperbolicity in the sense of Gromov of domains in R d (d ≥ 3) with respect to the minimal metric introduced by Forstnerič and Kalaj (Anal PDE 17(3):981–1003, 2024). In particular, we prove that every bounded strongly minimally convex domain is Gromov hyperbolic and its Gromov compactification is equivalent to its Euclidean closure. Moreover, we prove that the boundary of a Gromov hyperbolic convex domain does not contain non-trivial conformal harmonic disks. Finally, we study the relation between the minimal metric and the Hilbert metric in convex domains. [ABSTRACT FROM AUTHOR]

This study focuses on investigating the relativistic quantum dynamics of a fermionic oscillator field in the background of a minimal surface, specifically a catenoid bridge. Our primary objective is to analyze the bound-state solutions of this system by solving the non-perturbative radial wave equation. To achieve this, we consider two distinct scenarios. First, we explore the case where the particle is located inside the catenoid bridge and search for solutions to the wave equation. Subsequently, we extend our analysis to scenarios where the particle is positioned at a significant distance from the origin, comparable to the size of the catenoid bridge. In this regime, we obtain solutions to the wave equation to understand the behavior of the fermionic oscillator field. By examining these solutions, we investigate the influence of both the oscillator coupling and the spacetime background on the system. [ABSTRACT FROM AUTHOR]

This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We focus on the number of omitted values and the total weight of the totally ramified values of their Gauss maps. In particular, we construct new complete minimal surfaces of finite total curvature whose Gauss maps have 2 omitted values and 1 totally ramified value of order 2, that is, the total weight of the totally ramified values of their Gauss maps are 5 / 2 (= 2.5) in Euclidean 3-space and Euclidean 4-space, respectively. Moreover we discuss several outstanding problems in this study. [ABSTRACT FROM AUTHOR]

We prove that every open Riemann surface M is the complex structure of a complete surface of constant mean curvature 1 (CMC ‒1) in the three-dimensional hyperbolic space ℍ 3 . We go further and establish a jet interpolation theorem for complete conformal CMC ‒1 immersions M → ℍ 3 . As a consequence, we show the existence of complete densely immersed CMC ‒1 surfaces in ℍ 3 with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in ℂ 2 × ℂ ∗ which is also established in this paper. [ABSTRACT FROM AUTHOR]

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework. [ABSTRACT FROM AUTHOR]

We consider smooth Riemannian surfaces whose curvature K satisfies the relation Δ log | K - c | = a K + b away from points where K = c for some (a , b , c) ∈ R 3 , which we call generalized Ricci surfaces. We prove some isometric immersion theorems allowing points where K = c using properties of log-harmonic functions. For instance, we obtain a characterization of Riemannian surfaces that locally admit minimal isometric immersions, possibly with umbilical points, into a 3-dimensional Riemannian manifold of constant sectional curvature. We also give an application to convex affine spheres. Finally, we study compact generalized Ricci surfaces: we obtain topological obstructions and construct examples. [ABSTRACT FROM AUTHOR]

We show that only pseudo-spherical surface whose caustic becomes a minimal surface is Dini surface family. Moreover, we give the Weierstrass data for corresponding minimal surface to the caustic. [ABSTRACT FROM AUTHOR]

We explicitly characterize boundary conditions that are compatible with low order variational principles. The freedom afforded by adding in a null Lagrangian without altering the Euler–Lagrange equation significantly expands the range of variationally admissible boundary conditions, although not all possibilities are permitted. Applications to several fundamental problems arising in elastostatics, including bars, beams, and plates, are presented. [ABSTRACT FROM AUTHOR]

Toward identifying the number of minimal surfaces sharing the same boundary from the geometry of the boundary, we propose a numerical scheme with high speed and high accuracy. Our numerical scheme is based on the method of fundamental solutions. We establish the convergence analysis for Dirichlet energy and L ∞ -error analysis for mean curvature. Each of the approximate solutions in our scheme is a smooth surface, a significant difference from previous studies that required mesh generation. [ABSTRACT FROM AUTHOR]

We are interested in the solution of the Björling problem for timelike surfaces in R 1 4 \mathbb{R}_{1}^{4} . The main contribution of the paper is to present new and many examples of timelike zero mean curvature surfaces and give their explicit parametric equations. In particular cases, one observes that the parametric equations of these surfaces coincide with the timelike minimal surfaces in Lorentz–Minkowski 3-space. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate osculating type ruled surfaces with a type 2-Bishop frame in E 3 . We accomplish this by employing the symmetry of osculating curves. We examine osculating type ruled surfaces by taking into account the curvatures of the base curve. We investigate the geometric properties of these surfaces, focusing on their cylindrical and developable characteristics. Moreover, we calculate the Gaussian and mean curvatures and provide the requirements for the surface to be flat and minimal. We determine the requirements for the curves lying on this surface to be geodesic, asymptotic curves, or lines of curvature. Furthermore, relations between osculating type ruled surfaces with central tangent and central normal vectors are given. Finally, some examples of these surfaces are presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the various aspects of normal hyperbolic mean curvature flow by LeFloch and Smoczyk. It is remarkable that the equation admits the null condition in three-dimensional case and only satisfies the first null condition when n = 2. Based on the interesting findings, we can obtain the results of global existence of smooth solutions, as well as the stability of hyperplanes under this flow when 2 ≤ n ≤ 7 , which relates to the famous Bernstein theorem. Some explicit solutions for this flow have been also derived. It should be emphasized that the null structures of this hyperbolic mean curvature flow have not been found before. [ABSTRACT FROM AUTHOR]

In this paper we consider determining a minimal surface embedded in a Riemannian manifold Σ × R. We show that if Σ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine Σ up to an isometry. [ABSTRACT FROM AUTHOR]

We study geometric properties of caustics of pseudo-spherical surfaces, that is, surfaces with constant negative Gaussian curvature -1 in the Euclidean 3-space ℝ 3 . We investigate the Gaussian and the mean curvature of caustics of pseudo-spherical surfaces. Moreover, a certain condition required for the caustics to be minimal surfaces is derived. [ABSTRACT FROM AUTHOR]

This article describes a parametric design and fabrication workflow influenced by Frei Otto’s form-finding experiments on soap films. The research investigates minimal surface geometry by combining physical and digital experiments in a computational framework. Operating on mesh topology, various parametric design tools and plug-ins in Rhinoceros/Grasshopper are presented to discuss the translation of minimal surfaces to flat strips suitable for planar fabrication using flexible materials. These tools are tested on a case study to show the automated design and manufacture of double-curved surfaces as double-layered strips running in perpendicular directions that can be affixed at point connections for structural stability. The development of the parametric workflow, material constraints, and stripped fabrication of layers are discussed.

In this work, the study was focused on how deformable spacetime gives rise to physical laws, particularly quantum and special relativity. It was shown that both the Lorentz factor and quantum mechanics can emerge from the dynamical behavior of minimal surfaces in deformable spacetime, where the Möbius transformations prevail rather than the Lorentz transformations. The motion of deformable surfaces under minimality conditions creates a helicoidal pattern, a minimal surface with a metric designated by AdS (anti-de Sitter) space. Its isometry is a catenoid of which metric belongs to dS (de Sitter) space. When the two ends of the helicoid join it becomes a Möbius strip with multi-twists (i.e. windings), the so-called Möbius helicoid. It represents the most elementary mass with the attributes of charge and spin. It was shown that the number of windings increases with velocity according to the Lorentz factor. In other words, the Lorentz factor emerged from the transformation equations between the helicoid and catenoid, and both of them are isometric minimal surfaces. This fact serves as a means of transforming the metrics of the two isometric surfaces in AdS and dS spaces. There is an intimate relation between helicoidal motion, Möbius helicoid, and the Lorentz transformations. The zitterbewegung behavior of Dirac electron can be explained by the internal dynamics of the Möbius helicoid. Instead of using electromagnetic theory which led to the birth of special relativity, one can use the electrical circuit theory for deformable surface dynamics. A master equation was derived by using the resistive, capacitive, and inductive properties of deformable spacetime. The electromagnetic wave, Schrödinger, Klein–Gordon, Dirac, quantum telegrapher, and quantum torsion equations could be easily obtained from this master equation by making some simplifying assumptions for the parameters involved. The existing torsion due to helicoidal motion modifies the energy/mass equation (i.e.E=mc2) of special relativity. The Higgs potential can be obtained from the dynamic potential term of the master equation. Deformable space–time allows the emergence of a rich number of physical laws and relations. The masses of particles were calculated using the Möbius helicoid model for the very elementary structure of mass. The mass of a particle can be visualized as a three-dimensional network of helicoids with relatively higher stability. The stability is particularly associated with the phase changes of the catenoidal structure of the Möbius helicoid and the torsional vibrational modes of windings. Both the phase (or shape) change and/or the increase of the vibrational energies of windings produce a new configuration that represents a new particle. The masses of 187 particles were determined from this mechanism with an accuracy of ±3 MeV. It turned out that excitation energy states which could be expressed in terms of prime number multiples of a precursor mass play an important role in providing relatively higher stability.

Laminar flow offers significant potential for increasing the energy-efficiency of future transport aircraft. The German Cluster of Excellence SE2A is developing a new approach for hybrid laminar flow control. The concept aims to maintain laminar flow up to 80% of the chord length by integrating suction panels at the rear part of the wing, which consist of a thin suction skin and a supporting core structure. This study examines effects of various suction panel configurations on wing mass and load transfer for an all-electric short-range aircraft. Suction panel material, as well as thickness and relative density of the suction panel core are modified in meaningful boundaries. Suction panels made from Ti6Al4V offer the most robust design resulting in a significant increase in wing mass. For the studied configurations, they represent up to 33.8% of the mass of the wingbox. In contrast, panels made from Nylon11CF or PU1000 do not significantly increase the wing mass. However, the use of these materials raises questions about their robustness under operational conditions. The results demonstrate that the choice of material strongly influences the load path within the wing structure. Ti6Al4V suction panels provide sufficient mechanical properties to significantly contribute to load transfer and buckling stiffness. Locally, the share of load transfer attributed to the suction panel exceeds 50%. In contrast, compliant materials such as Nylon11CF or PU1000 are inherently decoupled from load transfer. Unlike the thickness of the suction skin, the relative density of the core structure strongly affects the wrinkling stiffness. However, wrinkling failure did not appear critical for the examined suction panel configurations. In the present study, the mechanical properties of Ti6Al4V cannot fully be exploited. Therefore, compliant suction panels made from Nylon11CF are preferred in order to achieve a lightweight solution, provided that they meet operational requirements. [ABSTRACT FROM AUTHOR]

We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in R³ with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least 2, for holomorphic null immersions into Cn with n≥3, for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for super-minimal immersions into any self-dual or anti-self-dual Einstein four-manifold. [ABSTRACT FROM AUTHOR]

This paper is devoted to the development of Beckenbach's theory of the meromorphic minimal surfaces. We consider the relationship between the number of separated maximum points of a meromorphic minimal surface and the Baernstein's T*-function. The results of Edrei, Goldberg, Heins, Ostrovskii, Wiman are generalized. We also give examples showing that the obtained estimates are sharp. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate osculating type ruled surfaces with a type 2-Bishop frame in E3. We accomplish this by employing the symmetry of osculating curves. We examine osculating type ruled surfaces by taking into account the curvatures of the base curve. We investigate the geometric properties of these surfaces, focusing on their cylindrical and developable characteristics. Moreover, we calculate the Gaussian and mean curvatures and provide the requirements for the surface to be flat and minimal. We determine the requirements for the curves lying on this surface to be geodesic, asymptotic curves, or lines of curvature. Furthermore, relations between osculating type ruled surfaces with central tangent and central normal vectors are given. Finally, some examples of these surfaces are presented.

In this study, we focus on Cartan null and pseudo null curves in Minkowski 3-space E 1 3 . Firstly we define Cartan null and pseudo null equivalent curves and give the related examples. Then we give a construction method for Cartan null curves and it is shown that every Cartan null curve can be obtained from a timelike curve lying in the Lorentz plane in E 1 3 . Also a construction method is given for pseudo null curves and we show that all pseudo null curves lie in a lightlike plane in E 1 3 . Finally we define a new surface in E 1 3 called “surface of pseudo null curves” and we obtain the geometric properties of such surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider rotational surfaces in the Euclidean 4-space E4 with the profile curve contained in a 2-plane. We obtain some classification results of such surfaces in terms of having generalized 1-type Gauss map. [ABSTRACT FROM AUTHOR]

We show that if E is a closed convex set in C n (n > 1) contained in a closed halfspace H such that E ∩ b H is nonempty and bounded, then the concave domain Ω = C n \ E contains images of proper holomorphic maps f : X → C n from any Stein manifold X of dimension < n , with approximation of a given map on closed compact subsets of X. If in addition 2 dim X + 1 ≤ n then f can be chosen an embedding, and if 2 dim X = n , then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties. [ABSTRACT FROM AUTHOR]

An equivariant minimal surface in C H n is a minimal map of the Poincaré disc into C H n which intertwines two actions of the fundamental group of a closed surface Σ : a Fuchsian representation on the disc and an irreducible action by isometries on C H n . The moduli space of these can been studied by relating it to the nilpotent cone in each moduli space of PU(n, 1)-Higgs bundles over the conformal surface corresponding to the map. By providing a necessary condition for points on this nilpotent cone to be smooth this article shows that away from the points corresponding to branched minimal immersions or ±-holomorphic immersions the moduli space is smooth. The argument is easily adapted to show that for R H n the full space of (unbranched) immersions is smooth. For C H 2 we show that the connected components of the moduli space of minimal immersions are indexed by the Toledo invariant and the Euler number of the normal bundle of the immersion. This is achieved by studying the limit points of the C × -action on the nilpotent cone. It is shown that the limit points as t → 0 lead only to branched minimal immersions or ±-holomorphic immersions. In particular, the Euler number of the normal bundle can only jump by passing through branched minimal maps. [ABSTRACT FROM AUTHOR]

In this article, we prove a Carleman inequality on a product manifold M×RM\times {\mathbb{R}}. As applications, we prove that (1) a periodic harmonic function on R2{{\mathbb{R}}}^{2} that decays faster than all exponential rate in one direction must be constant 0, (2) a periodic minimal hypersurface in R3{{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rate in one direction must be a hyperplane, and (3) a periodic translator in R3{{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rates in one direction must be a translating hyperplane.

We discover a geometric property of the space of tensors of fixed multilinear (Tucker) rank. Namely, it is shown that real tensors of the fixed multilinear rank form a minimal submanifold of the Euclidean space of tensors endowed with the Frobenius inner product. We also establish the absence of local extrema for linear functionals restricted to the submanifold of rank-one tensors, finding application in statistics. [ABSTRACT FROM AUTHOR]

Area-minimizing unit vector fields on antipodally punctured unit 2-sphere are defined in Fabiano et al. (Manuscr Math 161(3-4):487–499, 2019). For these unit vector fields with even Poincaré indexes, we prove that the topological closure of their image coincides with the image of minimally immersed Klein bottles in T 1 S 2 . [ABSTRACT FROM AUTHOR]

The total mean curvature functional for submanifolds into the Riemannian product space $\mathbb{S}^n\times\mathbb{R}$ is considered and its first variational formula is presented. Later on, two second-order differential operators are defined and a nice integral inequality relating both of them is proved. Finally, we prove our main result: an integral inequality for closed stationary $\mathcal{H}$ -surfaces in $\mathbb{S}^n\times\mathbb{R}$ , characterizing the cases where the equality is attained. [ABSTRACT FROM AUTHOR]

We provide a new characterization of the catenoid and the Delaunay surface among constant mean curvature surfaces touching a one-parameter family of spheres along arcs in a 3-dimensional Euclidean space. Furthermore we prove a higher-dimensional analogue for constant mean curvature hypersurfaces. We also give a generalization of the classical theorem due to Joachimsthal. [ABSTRACT FROM AUTHOR]

In this paper, it is characterized the generating curves of minimal rotational surfaces in the de Sitter space S 1 3 as solutions of a variational problem. More exactly, it is proved that these curves are the critical points of a potential energy functional involving the distance to a given plane among all curves of S 1 2 with prescribed endpoints and fixed length. This extends the known Euler's result that asserts that the catenary is the generating curve of the catenoid. [ABSTRACT FROM AUTHOR]

Cellular membranes display an incredibly diverse range of shapes, both in the plasma membrane and at membrane bound organelles. These morphologies are intricately related to cellular functions, enabling and regulating fundamental membrane processes. However, the biophysical mechanisms at the origin of these complex geometries are not fully understood from the standpoint of membrane-protein coupling. In this study, we focused on a minimal model of helicoidal ramps representative of specialized endoplasmic reticulum compartments. Given a helicoidal membrane geometry, we asked what is the distribution of spontaneous curvature required to maintain this shape at mechanical equilibrium? Based on the Helfrich energy of elastic membranes with spontaneous curvature, we derived the shape equation for minimal surfaces, and applied it to helicoids. We showed the existence of switches in the sign of the spontaneous curvature associated with geometric variations of the membrane structures. Furthermore, for a prescribed gradient of spontaneous curvature along the exterior boundaries, we identified configurations of the helicoidal ramps that are confined between two infinitely large energy barriers. Overall our results suggest possible mechanisms for geometric control of helicoidal ramps in membrane organelles based on curvature-inducing proteins.

In this paper, we study on three kinds of spacelike helicoidal surfaces in Minkowski 4-space. First, we give an isometry between such helicoidal surfaces and rotational surfaces which is a kind of generalization of Bour's theorem in Minkowski 3-space to Minkowski 4-space. Then, we show that if the isometric pair of surfaces have same Gauss map, then they are hyperplanar and minimal. Also, we give the parametrizations of isometric pair of surfaces having same Gauss map. Finally, we present some examples via Wolfram Mathematica 10.4. [ABSTRACT FROM AUTHOR]

We discuss conservation laws for thin structures which could be modeled as a material minimal surface, i.e., a surface with zero mean curvatures. The models of an elastic membrane and micropolar (six-parameter) shell undergoing finite deformations are considered. We show that for a minimal surface, it is possible to formulate a conservation law similar to three-dimensional non-linear elasticity. It brings us a path-independent J -integral which could be used in mechanics of fracture. So, the class of minimal surfaces extends significantly a possible geometry of two-dimensional structures which possess conservation laws. [ABSTRACT FROM AUTHOR]

In this paper, we have described the B-Lift curve in Euclidean space as a curve obtained by combining the endpoints of the binormal vector of a unit speed curve. Subsequently, we have explored the Frenet frames of the B-Lift curves. Moreover, we have introduced the tangent, normal and binormal surfaces of the B-Lift curve and examined the geometric invariants of these surfaces. Finally, we have investigated the singularities of these surface and visualized the surfaces with MATLAB program. [ABSTRACT FROM AUTHOR]

Let M 3 be an initial data set for the Einstein–Maxwell equations with vanishing magnetic field and Σ 2 ⊂ M a two-sided compact embedded strictly stable minimal surface which locally maximizes the charged Hawking mass. Under suitable constrains on M, we establish a local rigidity result and conclude that there exist a neighborhood of Σ in M isometric to the Reissner-Nordström-Anti-de Sitter space. At the same time, we will deduce an estimate for area of Σ ⊂ M a two-sided compact embedded surface of genus g (Σ) which is locally area-minimizing. In the equality case, we prove that the induced metric on Σ has constant Gauss curvature and locally M splits along Σ . [ABSTRACT FROM AUTHOR]

We use the theory of mean-stable surfaces (stable minimal surfaces included) to explore the static Einstein–Maxwell space-time. We first prove that the zero set of the lapse function must be contained in the horizon boundary. Then, we explore some implications of it providing some results of the nonexistence of stable minimal surfaces in the interior of an electrostatic space, subject to certain initial-boundary data. We finish by proving that the ADM mass is bounded from above by the Hawking quasi-local mass with charge. [ABSTRACT FROM AUTHOR]

In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space E 3 is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same is true for maximal surfaces in the Minkowski 3-space L 3 . In this article, we introduce a new deformation family that continuously connects minimal surfaces in E 3 and maximal surfaces in L 3 , and prove a Krust-type theorem for this deformation family. This result induces Krust-type theorems for various important deformation families containing the associated family and the López-Ros deformation. Furthermore, minimal surfaces in the isotropic 3-space I 3 appear in the middle of the above deformation family. We also prove another type of Krust's theorem for this family, which implies that the graphness of such minimal surfaces in I 3 strongly affects the graphness of deformed surfaces. The results are proved based on the recent progress of planar harmonic mapping theory. [ABSTRACT FROM AUTHOR]

The logarithmic spiral is one of the most known curves with applications in engineering. We consider product of the logarithmic spiral with a real line and tensor product of two logarithmic spirals and investigate their minimality or flatness. The Vrănceanu surface, with c1 a logarithmic spiral, is also considered.

We prove that in a closed Riemannian manifold with dimension between 3 and 7, either there are minimal hypersurfaces with arbitrarily large area, or there exist uncountably many stable minimal hypersurfaces. Moreover, the latter case has a very pathological Cantor set structure which does not show up in certain manifolds. Among the applications, we prove that there exist minimal hypersurfaces with arbitrarily large area in analytic manifolds. In the proof, we use the Almgren-Pitts min-max theory proposed by Marques-Neves, the ideas developed by Song in his proof of Yau's conjecture, and the resolution of the generic multiplicity-one conjecture by Zhou. [ABSTRACT FROM AUTHOR]

An asymptotically flat manifold is the union of an almost flat region and a compact non flat region. If there exists a black hole, we will call the boundary of non flat region the entrance to the black hole. Also, we will call the distance from the entrance to the outermost minimal surface the depth. We use an integral norm of Ricci curvature to obtain positive lower bounds on the depth and the radius of the entrance. Also we obtain an upper bound of the number of ends from the integral norm of Ricci curvature. [ABSTRACT FROM AUTHOR]

Laminar flow offers significant potential for increasing the energy-efficiency of future transport aircraft. The German Cluster of Excellence SE2A is developing a new approach for hybrid laminar flow control. The concept aims to maintain laminar flow up to 80% of the chord length by integrating suction panels at the rear part of the wing, which consist of a thin suction skin and a supporting core structure. This study examines effects of various suction panel configurations on wing mass and load transfer for an all-electric short-range aircraft. Suction panel material, as well as thickness and relative density of the suction panel core are modified in meaningful boundaries. Suction panels made from Ti6Al4V offer the most robust design resulting in a significant increase in wing mass. For the studied configurations, they represent up to 33.8% of the mass of the wingbox. In contrast, panels made from Nylon11CF or PU1000 do not significantly increase the wing mass. However, the use of these materials raises questions about their robustness under operational conditions. The results demonstrate that the choice of material strongly influences the load path within the wing structure. Ti6Al4V suction panels provide sufficient mechanical properties to significantly contribute to load transfer and buckling stiffness. Locally, the share of load transfer attributed to the suction panel exceeds 50%. In contrast, compliant materials such as Nylon11CF or PU1000 are inherently decoupled from load transfer. Unlike the thickness of the suction skin, the relative density of the core structure strongly affects the wrinkling stiffness. However, wrinkling failure did not appear critical for the examined suction panel configurations. In the present study, the mechanical properties of Ti6Al4V cannot fully be exploited. Therefore, compliant suction panels made from Nylon11CF are preferred in order to achieve a lightweight solution, provided that they meet operational requirements.

In the present paper, we discuss the singular minimal surfaces in Euclidean $3-$space $\mathbb{R}^{3}$ which are minimal. Such a surface is nothing but a plane, a trivial outcome. However, a non-trivial outcome is obtained when we modify the usual condition of singular minimality by using a special semi-symmetric metric connection instead of the Levi-Civita connection on $\mathbb{R}^{3}$. With this new connection, we prove that, besides planes, the singular minimal surfaces which are minimal are the generalized cylinders, providing their explicit equations. A trivial outcome is observed when we use a special semi-symmetric non-metric connection. Furthermore, our discussion is adapted to the Lorentz-Minkowski 3-space.

The design of heat exchangers may change dramatically through the use of additive manufacturing (AM). Additive manufacturing, colloquially known as 3D printing, enables the production of monolithic metal bodies, devoid of contact resistance. The small volume of the exchanger, its lightness of weight, and the reduction of its production costs, compared to conventional methods, make the production of heat exchangers by AM methods conventional technologies. The review study presents a new look at the TPMS as a promising type of developed surface that can be used in the area of heat transfer. (Thus far, the only attractive option. The most important feature of additive manufacturing is the ability to print the geometry of theoretically any topography. Such a topography can be a minimal surface or its extended version—triply periodic minimal surface (TPMS). It was practically impossible to manufacture a TPMS-based heat exchanger with the method of producing a TPMS.) The issues related to the methods of additive manufacturing of metal products and the cycle of object preparation for printing were discussed, and the available publications presenting the results of CFD simulations and experimental tests of heat exchangers containing a TPMS in their construction were widely discussed. It has been noticed that the study of thermal-flow heat transfer with the use of TPMSs is a new area of research, and the number of publications in this field is very limited. The few data (mainly CFD simulations) show that the use of TPMSs causes, on the one hand, a several-fold increase in the number of Nu, and on the other hand, an increase in flow resistance. The use of TPMSs in heat exchangers can reduce their size by 60%. It is concluded that research should be carried out in order to optimize the size of the TPMS structure and its porosity so that the gains from the improved heat transfer compensate for the energy expenditure on the transport of the working fluid. It has been noticed that among the numerous types of TPMSs available for the construction of heat exchangers, practically, four types have been used thus far: primitive, gyroid, I-WP, and diamond. At the moment, the diamond structure seems to be the most promising in terms of its use in the construction of heat exchangers and heat sinks. It is required to conduct experimental research to verify the results of the CFD simulation. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of infinite many solutions for a class of prescribed mean curvature equation of the type - d i v ∇ u 1 + | ∇ u | 2 = f (u) , in Ω , u = 0 , on ∂ Ω , where Ω ⊂ R N (N ≥ 1) is a smooth bounded domain, by assuming different conditions on the nonlinearity f. [ABSTRACT FROM AUTHOR]

The Gauss map g of a surface Σ in R 4 takes its values in the Grassmannian of oriented 2-planes of R 4 : G + (2 , 4) . We give geometric criteria of stability for minimal surfaces in R 4 in terms of g. We show in particular that if the area of the Gauss map | g (Σ) | of a minimal surface is smaller than 2 π then the surface is stable by deformations which fix the boundary of the surface. This answers the question of Barbosa and Do Carmo (Math Z 173:13–28, 1980) in R 4 . [ABSTRACT FROM AUTHOR]

Design multiphase heterogeneous structures in order to provide multifunctional properties has many applications in the field of material design. In this research, a new method for construction multiscale heterogeneous microstructure is presented. Using statistical correlation function, specifically, two-point correlation functions, bicontinuous two-phase structure is constructed that has solid and void phases. In order to construction the heterogeneous media, an exponentially decreasing sine function is used as autocovariance function. Then based on Schwartz P minimal surface the porous media structure, is divided into two parts; porous solid phase and void phase. From the point of view of continuity, the phases are investigated and it is observed both phases of the constructed microstructure are connected throughout the media. Using this method it is possible to construct bicontinuous multiscale microstructures that are solid and void phase in coarse scale and the solid phase can be constructed as void and solid phase in fine scale. [ABSTRACT FROM AUTHOR]