Minimal surfaces in Flat Three-Dimensional Spaces

2002 ◽  
pp. 1-14 ◽  
Author(s):  
William H. Meeks III
Keyword(s):  
Minimal Surfaces ◽  
Three Dimensional

scholarly journals The Björling problem for minimal surfaces in a Lorentzian three-dimensional Lie group

2014 ◽  
Vol 195 (1) ◽  
pp. 95-110 ◽  
Author(s):  
Adriana A. Cintra ◽  
Francesco Mercuri ◽  
Irene I. Onnis
Keyword(s):  
Lie Group ◽  
Minimal Surfaces ◽  
Three Dimensional ◽  
Björling Problem

scholarly journals Ruled minimal surfaces in the three-dimensional Heisenberg group

2013 ◽  
Vol 261 (2) ◽  
pp. 477-496 ◽  
Author(s):  
Heayong Shin ◽  
Young Wook Kim ◽  
Sung-Eun Koh ◽  
Hyung Yong Lee ◽  
Seong-Deog Yang
Keyword(s):  
Heisenberg Group ◽  
Minimal Surfaces ◽  
Three Dimensional

scholarly journals Geometric convergence results for closed minimal surfaces via bubbling analysis

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Lucas Ambrozio ◽  
Reto Buzano ◽  
Alessandro Carlotto ◽  
Ben Sharp
Keyword(s):  
Minimal Surfaces ◽  
Sharp Estimate ◽  
Quantitative Description ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Global Character ◽  
Multiplicity One ◽  
Convergence Results ◽  
Geometric Convergence ◽  
Number Of Ends

AbstractWe present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus $$\gamma $$ γ is sequentially compact for any $$\gamma \ge 1$$ γ ≥ 1 . Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity $$m\ge 1$$ m ≥ 1 , away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process.

scholarly journals A loop group method for minimal surfaces in the three-dimensional Heisenberg group

2016 ◽  
Vol 20 (3) ◽  
pp. 409-448
Author(s):  
Josef F. Dorfmeister ◽  
Jun-Ichi Inoguchi ◽  
Shimpei Kobayashi
Keyword(s):  
Heisenberg Group ◽  
Minimal Surfaces ◽  
Three Dimensional ◽  
Loop Group ◽  
Group Method

Forced Convection Computational Fluid Dynamics Analysis of Architected and Three-Dimensional Printable Heat Sinks Based on Triply Periodic Minimal Surfaces

2020 ◽  
Vol 13 (2) ◽  
Author(s):  
Oraib Al-Ketan ◽  
Mohamed Ali ◽  
Mohamad Khalil ◽  
Reza Rowshan ◽  
Kamran A. Khan ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Heat Transfer Coefficient ◽  
Minimal Surfaces ◽  
Heat Sink ◽  
Three Dimensional ◽  
Heat Sinks ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic

Abstract The drive for small and compact electronic components with higher processing capabilities is limited by their ability to dissipate the associated heat generated during operations, and hence, more advanced heat sink designs are required. Recently, the emergence of additive manufacturing techniques facilitated the fabrication of complex structures and overcame the limitation of traditional techniques such as milling, drilling, and casting. Therefore, complex heat sink designs are now easily realizable. In this study, we propose a design procedure for mathematically realizable architected heat sinks and investigate their performance using the computational fluid dynamics (CFD) approach. The proposed heat sinks are mathematically designed with topologies based on triply periodic minimal surfaces (TPMSs). Three-dimensional CFD models are developed using the starccm+ platform for uniform heat sinks and topologically graded heat sinks to study the heat transfer performance in forced convection domains. The overall heat transfer coefficient, surface temperature, and pressure drop versus the input heat sources as well as the Reynolds number are used to evaluate the heat sink performance. Moreover, temperature contours and velocity streamlines were examined to analyze the fluid flow behavior within the heat sinks. Results showed that the tortuosity and channel complexity of the Diamond solid-networks heat sink result in a 32% increase in convective heat transfer coefficient compared with the Gyroid solid-network heat sink which has the comparable surface area under the examined flow conditions. This increase is at the expense of increased pressure drops which increases by the same percentage. In addition, it was found that expanding channel size along flow direction using the porosity grading approach results in significant pressure drop (27.6%), while the corresponding drop in convective heat transfer is less significant (15.7%). These results show the importance of employing functional grading in the design of heat sinks. Also, the manufacturability of the proposed designs was assessed using computerized tomography (CT) scan and scanning electron microscopy (SEM) imaging performed on metallic samples fabricated using powder bed fusion techniques. A visible number of internal manufacturing defects can affect the performance of the proposed heat sinks.

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