On minimal surfaces on two-step Carnot groups

For graph mappings constructed from contact mappings of arbitrary two-step Carnot groups, conditions for the correct formulation of the minimal surfaces problem are found. A suitable notion of the increment of the (sub-Riemannian) area functional is introduced, the differentiability of this functional is proved, and necessary minimality conditions for graph surfaces are deduced. These conditions are also expressed in terms of the sub-Riemannian mean curvature.

Metric Properties of Level Surfaces of Holder Mappings Defined on Two-Step Carnot Groups

Microstructures with minimal surfaces can be often found in natural porous architectures, where the surface tension minimizes the area. The triply periodic minimal surfaces (TPMS) [1] are an example of such microstructures. Compared with other porous structures, TPMS have three significant features: firstly, their geometries can be completely expressed via analytical functions; secondly, TPMS are periodic in three independent directions and thirdly, the mean curvature of TPMS is zero [2]. Transforming the TPMS-based unit cell into a lattice structure has particular usage in aerospace, nuclear energy, and biomedical applications where light weight, high stiffness, and temperature resistance are of critical importance. In the presented studies, the failure behavior of four typical TPMS structures (Primitive, Gyroid, Neovius, and IWP) under compression was studied using finite element analysis. Numerical modeling of the damage propagation and strength prediction was performed by removing the finite elements in which the appropriate damage criterion is reached. Utilizing the equations of the generated TPMS structures, the wall thickness of unit cell was considered the main parameter that defined the ceramics volume fraction and should be taken into consideration. Therefore, various unit cell models for different wall thicknesses were generated and used to investigate the impact of the cell geometry on the damage initiation, propagation, and overall compression strength. The results of compression strength and damage development were compared with those of other TPMS structures for the same wall thickness and volume fraction. Finally, the grade TPMS porous structure was provided to verify the effect of wall thickness variation on damage evolution on the macroscale.

The desire for new mostly fluent “soft spaces” in architecture cannot be overseen any more. Therefore new tools and approaches are tried out to create architecture with special spatial qualities. In combination with common building-technologies and considered as architectural elements, spatially curved membrane structures and especially anticlastic Minimal Surfaces offer one possible approach to this topic. This paper presents the result of the research on Minimal Surfaces of membrane corner solutions representing one part of a comprehensive basic research in this field. In connection with the exploration of corner solutions, the meaning and potential of analog and digital approaches became obvious. Depending on the potential of the selected tool, the desired results or the purpose of the research one of these strategies will be preferred. Further on this paper partly reveals new correlations between Minimal Surface and boundary conditions and so far unknown rules in its selforganizing processes. Case studies document new capabilities in designing and creating space in architecture. Latest approaches are dealing with alternative boundary conditions, case studies and with software implementation in terms of scripting found rules.

where Ai(p) and A2(p) are the principal curvatures of M at p. When H is constant, M is called a surface of constant mean curvature. A surface is said to have finite type if it is homeomorphic to a closed surface with a finite number of points removed. An important problem in classical differential geometry is the classification of properly embedded finite type surfaces M of constant mean curvature in R . If M is a closed embedded surface of constant mean curvature, then it follows from Alexandrov [1] that M must be a round sphere. The classical examples of properly embedded surfaces with zero mean curvature are the plane, the helicoid and the catenoid. Surfaces of zero mean curvature are usually called minimal surfaces. The remaining classical examples of properly embedded surfaces of constant mean curvature were found by Delaunay [4]. The Delaunay surfaces are surfaces of revolution. Recently Hoffman and Meeks [6, 7] have found examples of properly embedded minimal surfaces which are homeomorphic to closed surfaces of positive genus with 3 points removed. Callahan, Hoffman and Meeks [3] have found other examples with more ends. An annular end E of & properly embedded surface in R 3 is a properly embedded annulus E in M where E is homeomorphic to S x [0,1). When M has finite type, then every end of M is annular. Hoffman and Meeks have developed a theory to deal with global problems concerning the geometry of properly embedded minimal surfaces M and, in particular, they show that most annular ends of M converge at infinity in R to a flat plane or to the end of a catenoid. Recently N. Kapouleas [8] in his thesis has shown that for every positive integer k > 2, there exists a properly embedded surface M& of finite type with nonzero mean curvature and with k ends. He also has constructed highergenus examples. As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant mean curvature have a special geometry and play an important role in global theorems.

On the construction of minimal surfaces from geodesics

Area-stationary surfaces in the Heisenberg group [formula omitted

<div class="section abstract"><div class="htmlview paragraph">Advancements in additive manufacturing (AM) technology have enabled the use of Triply Periodic Minimal Surface (TPMS) lattice structures to integrate thermal and structural functions into a single component. These structures offer advantages such as weight reduction, compactness and enhanced heat dissipation, making them promising for automotive, aerospace and electronics applications. TPMS structures, characterized by zero mean curvature and periodic crystalline geometry, have recently gained significant research attention thanks to their potential in thermal management. Among various TPMS geometries, the gyroid and diamond structures stand out for their thermal and fluid dynamic performance. This study explores the influence of cell geometry, unit cell size, and wall thickness on the efficiency of TPMS-based heat exchangers, as these parameters are crucial for their technical feasibility. Using Computational Fluid Dynamics (CFD) simulations, a comparative analysis is conducted for a case study represented by a heat exchanger. The numerical approach relies on a steady-state Reynolds-Averaged Navier-Stokes (RANS) approach with the Reynolds Stress Transport (RST) Elliptic Blending model, while heat transfer is analyzed through the Conjugate Heat Transfer (CHT) technique. The results indicate that reducing the unit cell size enhances heat transfer but also increases pressure drop at a fixed flow rate. Similarly, increasing the wall thickness raises pressure losses, though its effect on heat transfer is minimal. Overall, the diamond structure outperforms the gyroid in both thermal efficiency and flow permeability, making it a more effective choice for TPMS-based heat exchangers. These findings offer valuable insights for optimizing TPMS geometries in high-performance heat transfer applications, guiding future research and industrial implementations.</div></div>

The transient 1D dimensionless conduction heat rate approach applied to 3D TPMS structures

If $$\alpha \in \mathbb R$$ α ∈ R , an $$\alpha $$ α -stationary surface in Euclidean space is a surface $$\Sigma $$ Σ whose mean curvature H satisfies $$H(p)=\alpha |p|^{-2} \langle \nu ,p\rangle $$ H ( p ) = α | p | - 2 ⟨ ν , p ⟩ , $$p\in \Sigma $$ p ∈ Σ . These surfaces generalize in dimension two a classical family of curves studied by Euler which are critical points of the moment of inertia of planar curves. In this paper we establish, via inversions, a one-to-one correspondence between $$\alpha $$ α -stationary surfaces and $$-(\alpha +4)$$ - ( α + 4 ) -stationary surfaces. In particular, there is a correspondence between $$-4$$ - 4 -stationary surfaces and minimal surfaces. Using this duality we give some results of uniqueness of $$-4$$ - 4 -stationary surfaces and we solve the Börling problem.

This chapter reviews complex structures on a manifold, then gives an elementary exposition of the complex structure induced on a surface by a Riemannian metric. In this way a complex structure is induced on any surface immersed into one of the space forms. Surfaces immersed into Möbius space inherit a complex structure. In all cases we use this structure to define a reduction of a moving frame to a unique frame associated to a given complex coordinate. Umbilic points do not hinder the existence of these frames, in contrast to the obstruction they can pose for the existence of second order frame fields. The Hopf invariant and the Hopf quadratic differential play a prominent role in the space forms as well as in Möbius geometry. Using the structure equations of the Hopf invariant h, the conformal factor e u , and the mean curvature H of such frames, we give an elementary description of the Lawson correspondence between minimal surfaces in Euclidean geometry and constant mean curvature equal to one (CMC 1) surfaces in hyperbolic geometry; and between minimal surfaces in spherical geometry and CMC surfaces in Euclidean geometry.

Geometric Flow Theory is cross fertilized by diverse elements coming from Pure Mathematic and Mathematical Physic, but its foundation is mainly based on Riemannian Geometry, as explained by M. Berger in a recent panoramic view of this discipline, its extension to complex manifolds, the Erich Kähler's Geometry, vaunted for its unabated vitality by J.P. Bourguignon, and Minimal Surface Theory. This paper would like to initiate seminal studies for applying intrinsic geometric flows in the framework of information geometry theory. More specifically, after having introduced Information metric deduced for Complex Auto‐Regressive (CAR) models from Fisher Matrix (Siegel Metric and Hyper‐Abelian Metric from Entropic Kähler Potential), we study asymptotic behavior of reflection coefficients of CAR models driven by intrinsic Information geometric Kähler‐Ricci and Calabi flows. These Information geometric flows can be used in different contexts to define distance between CAR models interpreted as geodesics of Entropy Manifold. We conclude with potential application of Intrinsic Geometric Flow on Gauss Map to transform Manifold of any dimension by mean of Generalized Weierstrass Formula introduced by Kenmotsu that can represent arbitrary surfaces with non‐vanishing mean curvature in terms of the mean curvature function and the Gauss map. One of the advantages of the generalized formulae is that they allow to construct a new class of deformations of surfaces by use of Intrinsic Geometric Flow on Gauss Map. We conclude with the Heat equation interpretation in the framework of Information Geometry.

Main properties of Al2O3 thin films deposited by magnetron sputtering of an Al2O3 ceramic target at different radio-frequency power and argon pressure and their passivation effect on p-type c-Si wafers

Bence, Merriman, and Osher proposed an algorithm for computing the motion a hypersurface by mean curvature in terms of solutions of the usual heat equation, continually reinitialized after short time steps. In this paper, applying some techniques of asymptotic analysis for the Allen--Cahn equation, we give a rate of convergence of their algorithm for the motion of a smooth and compact hypersurface by mean curvature. We also consider the special case of a circle evolving by curvature and show that our rate is optimal.

In this paper, we develop some basic analytic tools to study compactness properties of J-curves (i.e. pseudoholomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves by establishing an extrinsic monotonicity principle for nonnegative functions f satisfying Δf ≥ -c2f, we show that curves locally parametrized as a graph over a coordinate tangent plane have all derivatives a priori bounded in terms of curvature and ambient geometry, and we thus establish ϵ-regularity for the square length of their second fundamental forms. These results are all provided for J-curves either with or without Lagrangian boundary and hold in almost all Hermitian manifolds of arbitrary even dimension (i.e. Riemannian manifolds for which the almost complex structure is an isometry).