Neutral Deformation Modes of Minimal Surfaces

In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space \(\mathbb {E}^3\) 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 \(\mathbb {L}^3\). In this article, we introduce a new deformation family that continuously connects minimal surfaces in \(\mathbb {E}^3\) and maximal surfaces in \(\mathbb {L}^3\), 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 \(\mathbb {I}^3\) 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 \(\mathbb {I}^3\) strongly affects the graphness of deformed surfaces. The results are proved based on the recent progress of planar harmonic mapping theory.

This paper proposes a new active bonding mechanism that achieves rigid, reversible, and nonback-drivable coupling between modular mobile robots in a chain formation. The first merit of this interface lies in its ability to operate in three independent modes. In the drive mode, the motor torque is routed to drive the module. In the clamp mode, the motor torque is redirected toward an active joint that enables one module to rotate relative to its neighbors in the formation. In the neutral mode, the motor's rotation achieves alignment between the interface's components prior to the initiation of the drive and clamp modes. The second merit stems from the dual-rod slider rocker (DRSR) mechanism, which toggles between the interface's three modes of operation. The design details of the interface are presented, as well as the optimal kinematic synthesis and dynamic analysis of the DRSR mechanism. Simulation and experimental results validate the DRSR's unique kinematics, as well as the rigidity and the three operation modes of the docking interface.

This paper presents a new design for a docking interface that enables rigid, reversible and non-back-drivable coupling between robotic modules in a chain architecture. The distinctive merit of the proposed interface is exhibited in its ability to operate in three independent modes. In the drive mode, the motor torque is directed to drive the host module. In the neutral mode, the motor torque aligns the coupling elements prior to docking. In the clamp mode, the motor torque actuates the revolute joint resulting from this docking process, thus allowing one module to revolve relative to its neighbors in the formation. In this paper, an optimality analysis of the unique kinematic properties of the dual-rod slider-rocker (DRSR) mechanism which enables this tri-state operation is presented. This analysis is supplemented by simulations and experimental results that validate these optimal kinematics, as well as the rigidity and the three operation modes of the docking interface.

We define certain deformations between minimal and non-minimal constant mean curvature (CMC) surfaces in Euclidean space $E^3$ which preserve the Hopf differential. We prove that, given a CMC $H$ surface $f$, either minimal or not, and a fixed basepoint $z_0$ on this surface, there is a naturally defined family $f_h$, for all real $h$, of CMC $h$ surfaces that are tangent to $f$ at $z_0$, and which have the same Hopf differential. Given the classical Weierstrass data for a minimal surface, we give an explicit formula for the generalized Weierstrass data for the non-minimal surfaces $f_h$, and vice versa. As an application, we use this to give a well-defined dressing action on the class of minimal surfaces. In addition, we show that symmetries of certain types associated with the basepoint are preserved under the deformation, and this gives a canonical choice of basepoint for surfaces with symmetries. We use this to define new examples of non-minimal CMC surfaces naturally associated to known minimal surfaces with symmetries.

The bifurcation in a dynamical system with cylindrical symmetry dependent on several parameters is studied with reference to the Couette-Taylor problem. Points at which two neutral curves intersect (bifurcation points of codimension 2) corresponding to several independent neutral modes are found. In the neighborhood of the bifurcation points of codimension 2 the interaction of these modes can be described by a system of amplitude equations on the central manifold. If the neutral modes are nonrotationally symmetrical, there exist seven different resonance states that influence the cubic terms of the amplitude system. For the resonances Res 0 and Res 3 the results of calculating the intersection points are presented and the conditions under which stationary regimes exist and are stable are analyzed.

Minimal surfaces are ubiquitous in nature. Here, they are considered as geometric objects that bear a deformation content. By refining the resolution of the surface deformation gradient afforded by the polar decomposition theorem, we identify a bending content and a class of deformations that leave it unchanged. These are the bending-neutral deformations, fully characterized by an integrability condition; they preserve normals. We prove that: (i) every minimal surface is transformed into a minimal surface by a bending-neutral deformation; (ii) given two minimal surfaces with the same system of normals, there is a bending-neutral deformation that maps one into the other; and (iii) all minimal surfaces have indeed a universal bending content.

We consider complete minimal surfaces (c.m.s.’s) in R 3 {R^3} and their deformations. M 1 {M_1} is an ε \varepsilon -deformation of M 0 {M_0} if M 1 {M_1} is a graph over M 0 {M_0} in an ε \varepsilon tubular neighborhood of M 0 {M_0} and M 1 {M_1} is ε C 1 \varepsilon \;{C^1} -close to M 0 {M_0} . A minimal surface M M is isolated if all c.m.s.’s which are sufficiently small deformations of M M are congruent to M M . In this paper we construct an example of a nonisolated c.m.s. It is modelled on a 4 4 -punctured sphere and is of finite total curvature. On the other hand, we prove that a c.m.s. discovered by Meeks and Jorge, modelled on the sphere punctured at the fourth roots of unity, is isolated.

Minimal surfaces with planar curvature lines are classical geometric objects, having been studied since the late 19th century. In this paper, we revisit the subject from a different point of view. After calculating their metric functions using an analytical method, we recover the Weierstrass data, and give clean parametrizations for these surfaces. Then, using these parametrizations, we show that there exists a single continuous deformation between all minimal surfaces with planar curvature lines. In the process, we establish the existence of axial directions for these surfaces.

We consider complete minimal surfaces (c.m.s.'s) in R3 and their deformations.Mi is an e-deformation of Mo if Mx is a graph over Aio inane tubular neighborhood of Mo and Mi is e C1-close to Mo.A minimal surface M is isolated if all c.m.s.'s which are sufficiently small deformations of M are congruent to M.In this paper we construct an example of a nonisolated c.m.s.It is modelled on a 4-punctured sphere and is of finite total curvature.On the other hand, we prove that a c.m.s.discovered by Meeks and Jorge, modelled on the sphere punctured at the fourth roots of unity, is isolated.

The aim of this paper is to investigate a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. We study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that the well-known López–Ros deformation of minimal surfaces is a special case of this transformation. We express the simple factor dressing and the López–Ros deformation explicitly in terms of the minimal surface and its conjugate surface. In particular, we can control periods and end behaviour of the simple factor dressing. This allows to construct new examples of doubly-periodic minimal surfaces arising as simple factor dressings of Scherk’s first surface.

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construction. The parametrization uses Hitchin's abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1-dimensional torus. This description is used to develop a spectral curve theory for the Lawson surface. This theory applies as well to other CMC and minimal surfaces with the same holomorphic symmetries as the Lawson surface but different Riemann surface structure. Additionally, we study the space of isospectral deformations of compact minimal surface of genus $g\geq2$ and prove that it is generated by simple factor dressing.

Dynamic compressive behavior of functionally graded triply periodic minimal surface meta-structures

Historically, the creation of lightweight, yet mechanically robust, materials have been the most sought‐after engineering pursuit. For that purpose, research efforts are dedicated to finding pathways to emulate and mimic the resilience offered by natural biological systems (i.e., bone and wood). These natural systems evolved over time to provide the most attainable structural efficiency through their architectural characteristics that can span over multiple length scales. Nature‐inspired man‐made cellular metamaterials have effective properties that depend largely on their topology rather than composition and are hence remarkable candidates for a wide range of application. Despite their geometrical complexity, the fabrication of such metamaterials is made possible by the emergence of advanced fabrication techniques that permit the fabrication of complex architectures down to the nanometer scale. In this work, we report the fabrication and mechanical testing of nature‐inspired, mathematically created, micro‐architected, cellular metamaterials with topologies based on triply periodic minimal surfaces (TPMS) with cubic symmetries fabricated through direct laser writing two‐photon lithography. These TPMS‐based microlattices are sheet/shell‐ and strut‐based metamaterials with high geometrical complexity. Interestingly, results show that TPMS sheet‐based microlattices follow a stretching‐dominated mode of deformation, and further illustrate their mechanical superiority over the traditional and well‐known strut‐based microlattices and microlattice composites. The TPMS sheet‐based polymeric microlattices exhibited mechanical properties superior to other micrloattices comprising metal‐ and ceramic‐coated polymeric substrates and, interestingly, are less affected by the change in density, which opens the door for fabricating ultralightweight materials without much sacrificing mechanical properties.

In this study, impact behavior and deformation mechanism of triply periodic minimal surface (TPMS) sheet (Schwarz’ P, Schoen’s I-WP, and Neovius) were numerically investigated. The convergence study of mesh size, sensitivity of cell number, and parametric investigation of shell thickness were performed. Three different collapse modes “” “” and “Whole-X” were experienced corresponding to Schwarz’ P, Schoen’s I-WP, and Neovius sheets, respectively. Comparison between the anisotropic 2D circular honeycomb and isotropic 3D TPMS were conducted. The results show that TPMS sheets perform excellent mechanical properties due to their isotropic smooth surface transitions and open-cell connectivity. Highlights Cell number sensitivity of TPMS structures is investigated and Mesh size convergence study of TPMS structure is performed. Systematic regular TPMS patterns are obtained and collapse mechanisms related to deformation mode are revealed. Comparison between TPMS structure and the representative 2D cellular circular honeycomb structure is conducted, better performance in withstanding uniaxial compression is confirmed.

Triply periodic minimal surface (TPMS) structures are widely utilized in fields such as aerospace, biomedicine, and sound absorption. This study investigated the fabrication characteristics and compressive properties of 316L stainless steel TPMS structures produced via selective laser melting (SLM). Uniform and gradient Gyroid (G) and I-Wrapped Package (IWP) porous structures with different porosities and unit cell sizes were designed. Specimens were fabricated via SLM and their morphology was evaluated employing scanning electron microscopy (SEM) along with micro-computed tomography (Micro-CT). The Micro-CT and SEM images revealed no defects or broken cells, confirming that SLM can accurately manufacture TPMS porous structures. After fabrication, the actual porosities of uniform porous specimens were 1.62%–5.14% lower than their target values. Both the elastic modulus and yield strength of uniform porous structures decreased progressively with increasing porosity. Although the deformation mode of TPMS structures was unchanged by graded porosity, introducing a porosity gradient significantly increased the compressive strength and energy absorption compared to uniform porosity designs. Using Abaqus software, a finite element model was established to analyze compressive behavior, enabling simulation of the corresponding stress–strain response. The outcomes of the numerical analysis demonstrated strong agreement with the experimental observations. This study systematically compares and validates the uniform porosity and gradient porosity in the 316L TPMS lattice through experiments and finite element analysis. These findings offer guidelines for designing TPMS metamaterials tailored to various operating conditions.