scholarly journals Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

2015 ◽  
Vol 111 (4) ◽  
pp. 851-886 ◽  
Author(s):  
A. Alarcón ◽  
B. Drinovec Drnovšek ◽  
F. Forstnerič ◽  
F. J. López
Keyword(s):  
Riemann Surface ◽  
Minimal Surface ◽  
Jordan Curves ◽  
Conformal Minimal Surface

scholarly journals Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve

2018 ◽  
Vol 2018 (740) ◽  
pp. 77-109 ◽  
Author(s):  
Antonio Alarcón ◽  
Franc Forstnerič
Keyword(s):  
Riemann Surface ◽  
Minimal Surface ◽  
Minimal Immersion ◽  
Open Riemann Surface ◽  
The Real ◽  
Conformal Minimal Immersion ◽  
Null Curve ◽  
Minimal Immersions ◽  
Conformal Minimal Surface

Abstract We show that for every conformal minimal immersion {u:M\to\mathbb{R}^{3}} from an open Riemann surface M to {\mathbb{R}^{3}} there exists a smooth isotopy {u_{t}:M\to\mathbb{R}^{3}} ( {t\in[0,1]} ) of conformal minimal immersions, with {u_{0}=u} , such that {u_{1}} is the real part of a holomorphic null curve {M\to\mathbb{C}^{3}} (i.e. {u_{1}} has vanishing flux). If furthermore u is nonflat, then {u_{1}} can be chosen to have any prescribed flux and to be complete.

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 Local minima of the Gauss curvature of a minimal surface

1991 ◽  
Vol 44 (3) ◽  
pp. 397-404
Author(s):  
Shinji Yamashita
Keyword(s):  
Minimal Surface ◽  
Gauss Curvature ◽  
Gauss Map ◽  
Local Minima ◽  
Jordan Curves ◽  
The Third ◽  
Isolated Points ◽  
Set Of Points

Let D be a domain in the complex ω-plane and let x: D → R3 be a regular minimal surface. Let M(K) be the set of points ω0 ∈ D where the Gauss curvature K attains local minima: K(ω0) ≤ K(ω) for |ω – ω0| < δ(ω0), δ(ω0) < 0. The components of M(K) are of three types: isolated points; simple analytic arcs terminating nowhere in D; analytic Jordan curves in D. Components of the third type are related to the Gauss map.

scholarly journals Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

2017 ◽  
Vol 28 (09) ◽  
pp. 1740004 ◽  
Author(s):  
Antonio Alarcón ◽  
Finnur Lárusson
Keyword(s):  
Riemann Surface ◽  
Convex Hull ◽  
Minimal Surface ◽  
Cohomology Class ◽  
Holomorphic Maps ◽  
De Rham Cohomology ◽  
Surface Theory ◽  
Open Riemann Surface ◽  
De Rham ◽  
Smooth Locus

Let [Formula: see text] be a connected open Riemann surface. Let [Formula: see text] be an Oka domain in the smooth locus of an analytic subvariety of [Formula: see text], [Formula: see text], such that the convex hull of [Formula: see text] is all of [Formula: see text]. Let [Formula: see text] be the space of nondegenerate holomorphic maps [Formula: see text]. Take a holomorphic 1-form [Formula: see text] on [Formula: see text], not identically zero, and let [Formula: see text] send a map [Formula: see text] to the cohomology class of [Formula: see text]. Our main theorem states that [Formula: see text] is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on [Formula: see text] can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

scholarly journals Every Meromorphic Function is the Gauss Map of a Conformal Minimal Surface

2017 ◽  
Vol 29 (4) ◽  
pp. 3011-3038 ◽  
Author(s):  
A. Alarcón ◽  
F. Forstneric̆ ◽  
F. J. López
Keyword(s):  
Meromorphic Function ◽  
Minimal Surface ◽  
Gauss Map ◽  
Conformal Minimal Surface

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Visualization of Enneper’s surface by line graphics

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 387-405 ◽  
Author(s):  
Vesna Velickovic
Keyword(s):  
Minimal Surface ◽  
Graphical Representations ◽  
Lines Of Curvature ◽  
Geodesic Lines

Here we study Enneper?s minimal surface and some of its properties. We compute and visualize the lines of self-intersection, lines of intersections with planes, lines of curvature, asymptotic and geodesic lines of Enneper?s surface. For the graphical representations of all the results we use our own software for line graphics.

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