SINGULARITY KNOTS OF MINIMAL SURFACES IN ℝ4

2011 ◽  
Vol 20 (04) ◽  
pp. 513-546 ◽  
Author(s):  
MARC SORET ◽  
MARINA VILLE
Keyword(s):  
Minimal Surface ◽  
Branch Point ◽  
Minimal Surfaces ◽  
Crossing Number ◽  
Alexander Polynomial

We study knots in 𝕊3 obtained by the intersection of a minimal surface in ℝ4 with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

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 Lower Bounds on Virtual Crossing Number and Minimal Surface Genus

2011 ◽  
pp. 31-43 ◽  
Author(s):  
Kumud Bhandari ◽  
H. A. Dye ◽  
Louis H. Kauffman
Keyword(s):  
Minimal Surface ◽  
Lower Bounds ◽  
Crossing Number

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.

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.

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.

Periodic minimal surfaces embedded in ℝ3 derived from the singly periodic Scherk minimal surface

2018 ◽  
Vol 22 (01) ◽  
pp. 1850075
Author(s):  
Filippo Morabito
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Parameter Family ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic

We construct three kinds of periodic minimal surfaces embedded in [Formula: see text] We show the existence of a [Formula: see text]-parameter family of minimal surfaces invariant under the action of a translation by [Formula: see text] which seen from a distance look like [Formula: see text] equidistant parallel planes intersecting orthogonally [Formula: see text] equidistant parallel planes, [Formula: see text] [Formula: see text] We also consider the case where the surfaces are asymptotic to [Formula: see text] equidistant parallel planes intersecting orthogonally infinitely many equidistant parallel planes. In this case, the minimal surfaces are doubly periodic, precisely they are invariant under the action of two orthogonal translations. Last we construct triply periodic minimal surfaces which are invariant under the action of three orthogonal translations in the case of two stacks of infinitely many equidistant parallel planes which intersect orthogonally.

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