The classical theory of minimal surfaces

Meeks, William H.; Pérez, Joaquín

doi:10.1090/s0273-0979-2011-01334-9

Cited by 72 publications

(69 citation statements)

References 166 publications

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“…Therefore one obtains the requirement 28) which justifies the arguments used in [11,27]. If we focus on minimal surfaces, which by definition satisfy condition (2.4) then of the six invariants involved in (2.26) only five are non-trivial.…”

Section: Jhep07(2015)156mentioning

confidence: 75%

“…[28] for different (but equivalent) definitions and characterisations of minimality). We should note here that the mathematical terminology is somewhat (and uncharacteristically) imprecise, as these surfaces are called minimal surfaces regardless of whether or not the area is actually a minimum (and not some other extremum or critical point) of the area functional.…”

Section: Minimal Surfacesmentioning

confidence: 99%

“…It is always possible to introduce isothermal coordinates (ρ,φ) such that the induced metric is conformally flat, i.e. γρρ = γφφ, γρφ = 0 (see e.g [28,48]). In the case at hand, this can be done in a way compatible with the manifest isometry associated with the original φ direction, i.e.…”

Section: Jhep07(2015)156mentioning

confidence: 99%

“…We do not need to solve this explicitly, instead we note that if a surface, embedded in R 3 , is written in isothermal coordinates then it is minimal if X i (ρ, φ) for i = 1, 2, 3 is harmonic [28,48], i.e., if ∂ 2 ρ X i (ρ, φ)+∂ 2 φ X i (ρ, φ) = 0. This means that, assuming y(ρ), z(ρ), k(ρ) to satisfy (2.70), then from (2.68) we must havẽ…”

Section: Jhep07(2015)156mentioning

confidence: 99%

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Blackfolds, plane waves and minimal surfaces

Armas

Blau

2015

J. High Energ. Phys.

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Minimal surfaces in Euclidean space provide examples of possible non-compact horizon geometries and topologies in asymptotically flat space-time. On the other hand, the existence of limiting surfaces in the space-time provides a simple mechanism for making these configurations compact. Limiting surfaces appear naturally in a given space-time by making minimal surfaces rotate but they are also inherent to plane wave or de Sitter spacetimes in which case minimal surfaces can be static and compact. We use the blackfold approach in order to scan for possible black hole horizon geometries and topologies in asymptotically flat, plane wave and de Sitter space-times. In the process we uncover several new configurations, such as black helicoids and catenoids, some of which have an asymptotically flat counterpart. In particular, we find that the ultraspinning regime of singly-spinning Myers-Perry black holes, described in terms of the simplest minimal surface (the plane), can be obtained as a limit of a black helicoid, suggesting that these two families of black holes are connected. We also show that minimal surfaces embedded in spheres rather than Euclidean space can be used to construct static compact horizons in asymptotically de Sitter space-times.

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Section: Jhep07(2015)156mentioning

confidence: 75%

Section: Minimal Surfacesmentioning

confidence: 99%

Section: Jhep07(2015)156mentioning

confidence: 99%

Section: Jhep07(2015)156mentioning

confidence: 99%

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Blackfolds, plane waves and minimal surfaces

Armas

Blau

2015

J. High Energ. Phys.

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“…The foams obeyed Plateau's laws because of the requirement for minimal total surface area that has been mathematically proven by Jean Taylor using the geometric measure theory. 31 Other configurations are unstable and tend to quickly rearrange the bubbles in the intermediate state during the foaming process into this type of structure. The matching to Plateau's laws strongly evidenced that the designed foaming process as shown in Figure 1a indeed occurred and resulted in these 3D WG foams.…”

Section: Resultsmentioning

confidence: 99%

3D white graphene foam scavengers: vesicant-assisted foaming boosts the gram-level yield and forms hierarchical pores for superstrong pollutant removal applications

2015

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Three-dimensional (3D) nanostructures assembled with one-or few-layered ultrathin two-dimensional (2D) crystals have triggered great interest in energy and environmental applications. Here, we introduce a gas-foaming process in an hexagonal boron nitride (h-BN) ceramic material to fabricate 3D white graphene (WG) foams without using any catalysts or templates for superstrong pollutant removal applications. Importantly, the introduction of vesicants guaranteed the reproducibility and yield (4500 cm 3 ). Interestingly, these 3D WG foams possessed a vesicular structure with hierarchical pores ranging from nm to μm scales and with ultrathin walls consisting of mono-or few-layered BN membranes with planar sizes as large as 100 μm. Consequently, such microstructure merits of hierarchical pores and ultrathin walls endowed them not only very low density (2.1 mg cm − 3 ) but also superstrong adsorption ability, illustrated by capacitances up to 190 times its own weight toward a wide range of environment contaminations, including various oils and dyes. Thus, the 3D h-BN WG foams prepared by vesicant-assisted foaming should have great potential as outstanding environmental scavengers. NPG Asia Materials (2015) 7, e168; doi:10.1038/am.2015.8; published online 27 March 2015 INTRODUCTION Two-dimensional (2D) crystals, such as graphene 1 and white graphene (WG, mono-or few-layered hexagonal boron nitride (h-BN)), 2,3 have triggered great interest because of their extraordinary intrinsic properties and wide range of applications in electronics, optoelectronics, energy storage and the environment. 4 However, for some specific applications, such as the adsorption of various contaminants and as electrodes in electrochemical cells, their pristine flat 2D structures have been recognized to not fully match the practical requirements. [5][6][7][8] In contrast, three-dimensional (3D) architectures using 2D crystals as building blocks can simultaneously provide the virtues of 2D and 3D structures, such as ultrathin sheets and large specific surface areas from 2D sheets 6 and hierarchical pores and ultralight densities from 3D configurations. 7 Recently, such novel 2D-3D structure features have been proven to exhibit new and outstanding performances. For instance, graphenecarbon nanotube 3D structures had densities as low as 0.16 mg cm − 3 , even lighter than air (1.29 mg cm − 3 ); 9 3D graphene and BN networks exhibited excellent mechanical properties; 10,11 3D BNC hybrid networks showed tunable electronic and thermal properties. 12 However, the high-yield fabrication of such 3D architectures of 2D crystals, especially without using any templates or catalysts, remains a great challenge. Currently, there are two methods for fabricating 3D

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Ultralight Membrane Structures Toward a Sustainable Environment

Comitti,

Vijayakumaran,

Nejabatmeimandi

et al. 2024

Sustainable Structures and Buildings

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The building construction industry is the largest anthropogenic source of pollution, with massive energy consumption and substantial CO2 emissions. Lightweight tension structures allow the simultaneous implementation of several sustainable strategies by using recyclable low-carbon structural membranes offering a greener alternative to glass and other cladding materials. Their efficient structural load-bearing mechanisms result in significant weight savings in buildings and a drastic reduction of the environmental impact associated with material production, transportation, use, and disposal. A subgroup of lightweight materials, structural fabrics, and foils has been gaining popularity among designers and architects in recent years because of their desirable features such as high stiffness, strength, ductility, durability, and functional properties. While these structural membranes open new crucial perspectives for the clean energy transition and have been recently employed worldwide, their full potential is still limited by the lack of construction codes, advanced optimization tools, and comprehensive viscous-thermo-mechanical constitutive models. This chapter aims to foster the design of membrane structures by presenting their basic principles and recent advancements in the field. It covers the design approaches, employed materials and efforts in their characterization and modeling, implications on the sustainability of the built environment, current challenges, and future pathways from both academic research and engineering design viewpoints.

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The classical theory of minimal surfaces

Cited by 72 publications

References 166 publications

Blackfolds, plane waves and minimal surfaces

Blackfolds, plane waves and minimal surfaces

3D white graphene foam scavengers: vesicant-assisted foaming boosts the gram-level yield and forms hierarchical pores for superstrong pollutant removal applications

Ultralight Membrane Structures Toward a Sustainable Environment

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