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.

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 Free boundary minimal surfaces in the unit ball with low cohomogeneity

2016 ◽  
Vol 145 (4) ◽  
pp. 1671-1683 ◽  
Author(s):  
Brian Freidin ◽  
Mamikon Gulian ◽  
Peter McGrath
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Minimal Surfaces

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).

Approximation of minimal surfaces with free boundaries: convergence results

2018 ◽  
Vol 39 (3) ◽  
pp. 1391-1420
Author(s):  
Tristan Jenschke
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Minimal Surfaces ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Free Boundaries ◽  
Convergence Estimate ◽  
Convergence Results ◽  
Discrete Finite Element

Abstract In a previous paper we developed a penalty method to approximate solutions of the free boundary problem for minimal surfaces by solutions of certain variational problems depending on a parameter $\lambda $. There we showed existence and $C^2$-regularity of these solutions as well as convergence to the solution of the free boundary problem for $\lambda \to \infty $. In this paper we develop a fully discrete finite element procedure for approximating solutions of these variational problems and prove a convergence estimate, which includes an order of convergence with respect to the grid size.

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.

Optimal boundary regularity for minimal surfaces with a free boundary

1981 ◽  
Vol 33 (3-4) ◽  
pp. 357-364 ◽  
Author(s):  
Stefan Hildebrandt ◽  
Johannes C. C. Nitsche
Keyword(s):  
Free Boundary ◽  
Minimal Surfaces ◽  
Boundary Regularity ◽  
Optimal Boundary
Sign in / Sign up

Export Citation Format

Share Document