Polar varieties and bipolar surfaces of minimal surfaces in the n-sphere

2021 ◽  
Author(s):  
Katsuhiro Moriya
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Minimal Immersion ◽  
The Other ◽  
Branch Points

AbstractFor a given minimal surface in the n-sphere, two ways to construct a minimal surface in the m-sphere are given. One way constructs a minimal immersion. The other way constructs a minimal immersion which may have branch points. The branch points occur exactly at each point where the original minimal surface is geodesic. If a minimal surface in the 3-sphere is given, then these ways construct Lawson’s polar variety and bipolar surface.

Minimal surfaces with self-intersections along straight lines. I. Derivation and properties

1999 ◽  
Vol 55 (1) ◽  
pp. 58-64 ◽  
Author(s):  
E. Koch ◽  
W. Fischer
Keyword(s):  
Minimal Surface ◽  
Symmetry Group ◽  
Minimal Surfaces ◽  
Special Kind ◽  
Branch Points ◽  
Euler Characteristics ◽  
Oriented Surface ◽  
Periodic Minimal Surface ◽  
Full Symmetry ◽  
Straight Lines

A special kind of three-periodic minimal surface has been studied, namely surfaces that are generated from disc-like-spanned skew polygons and that intersect themselves exclusively along straight lines. A new procedure for their derivation is introduced in this paper. Several properties of each such surface may be deduced from its generating polygon: the full symmetry group of the surface, its orientability, the symmetry group of the oriented surface, the pattern of self-intersections, the branch points of the surface, the symmetry and periodicity of the spatial subunits demarcated by the surface, and the Euler characteristics both of the surface and of the spatial subunits. The corresponding procedures are described and illustrated by examples.

Description of a 3-periodic minimal surface family with trigonal symmetry

1994 ◽  
Vol 209 (1) ◽  
Author(s):  
A. Fogden
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Systematic Analysis ◽  
Trigonal Symmetry ◽  
Triply Periodic Minimal Surfaces ◽  
Periodic Minimal Surface ◽  
Triply Periodic ◽  
The Family

AbstractA systematic analysis of a family of triply periodic minimal surfaces of genus seven and trigonal symmetry is given. The family is found to contain five such surfaces free from self-intersections, three of which are previously unknown. Exact parametrisations of all surfaces are provided using the Weierstrass representation.

scholarly journals On solving the plateau problem in parametric form

1983 ◽  
Vol 6 (2) ◽  
pp. 341-361
Author(s):  
Baruch cahlon ◽  
Alan D. Solomon ◽  
Louis J. Nachman
Keyword(s):  
Minimal Surface ◽  
Numerical Method ◽  
Minimal Surfaces ◽  
Plateau Problem ◽  
Parametric Form ◽  
First Variation ◽  
Plateau’S Problem ◽  
Numerical Example ◽  
Plateau's Problem ◽  
The First Variation

This paper presents a numerical method for finding the solution of Plateau's problem in parametric form. Using the properties of minimal surfaces we succeded in transferring the problem of finding the minimal surface to a problem of minimizing a functional over a class of scalar functions. A numerical method of minimizing a functional using the first variation is presented and convergence is proven. A numerical example is given.

Geodesic Groups of Minimal Surfaces

1958 ◽  
Vol 10 ◽  
pp. 89-96
Author(s):  
H. G. Helfenstein
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces

In a previous paper (6) we have studied those minimal surfaces which admit geodesic mappings without isometries or similarities on another, not necessarily minimal, surface. Here we determine all pairs of minimal surfaces which can be geodesically mapped on each other. We find that two such surfaces are either: (i) similar Bonnet associates of each other, or (ii) both Poisson surfaces (that is, isometric to a plane), or (iii) both Scherk surfaces (2).

scholarly journals Embedded minimal surfaces of finite topology

2019 ◽  
Vol 2019 (753) ◽  
pp. 159-191 ◽  
Author(s):  
William H. Meeks III ◽  
Joaquín Pérez
Keyword(s):  
Asymptotic Behavior ◽  
Riemann Surface ◽  
Finite Number ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Compact Riemann Surface ◽  
Finite Topology ◽  
Compact Boundary ◽  
Interior Points

AbstractIn this paper we prove that a complete, embedded minimal surface M in {\mathbb{R}^{3}} with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface {\overline{M}} with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion {\overline{M}}. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

scholarly journals C∞-Convergence of Circle Patterns to Minimal Surfaces

2009 ◽  
Vol 194 ◽  
pp. 149-167 ◽  
Author(s):  
Shi-Yi Lan ◽  
Dao-Qing Dai
Keyword(s):  
Minimal Surface ◽  
Existence Theorem ◽  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Circle Pattern ◽  
Vertex Set ◽  
Circle Patterns ◽  
Simply Connected Region ◽  
Simply Connected ◽  
The Relationship

AbstractGiven a smooth minimal surface F: Ω → ℝ3 defined on a simply connected region Ω in the complex plane ℂ, there is a regular SG circle pattern . By the Weierstrass representation of F and the existence theorem of SG circle patterns, there exists an associated SG circle pattern in ℂ with the combinatoric of . Based on the relationship between the circle pattern and the corresponding discrete minimal surface F∊: → ℝ3 defined on the vertex set of the graph of , we show that there exists a family of discrete minimal surface Γ∊: → ℝ3, which converges in C∞(Ω) to the minimal surface F: Ω → ℝ3 as ∊ → 0.

scholarly journals HELICOIDAL MINIMAL SURFACES IN ℍ2×ℝ

2011 ◽  
Vol 86 (1) ◽  
pp. 135-149 ◽  
Author(s):  
YOUNG WOOK KIM ◽  
SUNG-EUN KOH ◽  
HEAYONG SHIN ◽  
SEONG-DEOG YANG
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Parameter Group ◽  
Screw Motions ◽  
Conjugate Surfaces

AbstractIt is shown that a minimal surface in ℍ2×ℝ is invariant under a one-parameter group of screw motions if and only if it lies in the associate family of helicoids. It is also shown that the conjugate surfaces of the parabolic and hyperbolic helicoids in ℍ2×ℝ are certain types of catenoids.

scholarly journals Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

2018 ◽  
Vol 2020 (18) ◽  
pp. 5630-5641 ◽  
Author(s):  
Brian Freidin ◽  
Peter McGrath
Keyword(s):  
Free Boundary ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Geodesic Ball ◽  
Euclidean Setting

Abstract We prove that the area of a free boundary minimal surface $\Sigma ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $\Sigma $ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows works of Brendle and Fraser–Schoen in the euclidean setting.

scholarly journals Reverse complementary matches simultaneously promote both back-splicing and exon-skipping

2021 ◽  
Author(s):  
Dong Cao
Keyword(s):  
Larval Stage ◽  
Large Scale ◽  
Exon Skipping ◽  
The Other ◽  
Circular Rnas ◽  
Branch Points ◽  
Rna Seq ◽  
C Elegans ◽  
Neuronal Genes ◽  
Whole Transcriptome

Circular RNAs (circRNAs) play diverse roles in different biological and physiological environments. Understanding circRNA expression profile and how circRNAs are regulated help make better use of them. Here, using large-scale neuron isolation from the first larval stage (L1) of Caenorhabditis elegans (C. elegans) followed by whole-transcriptome RNA sequencing (RNA-seq), we provide the first neuronal circRNA data in C. elegans. We show that circRNAs are highly expressed in the neurons of C. elegans and are preferably derived from neuronal genes. More importantly, reverse complementary matches (RCMs) in circRNA-flanking introns are not only required for back-splicing but also promote the skipping of exon(s) to be circularized. Finally, through one-by-one mutagenesis of all the splicing sites and branch points required for exon-skipping and back-splicing, we show that exon-skipping is not absolutely required for back-splicing, neither the other way. Instead, the coupled exon-skipping and back-splicing are happening at the same time and the two pathways work together to regulate the levels of the circular and the skipped transcripts.

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