scholarly journals Applications of Quaternionic Holomorphic Geometry to minimal surfaces

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
K. Leschke ◽  
K. Moriya
Keyword(s):  
Minimal Surface ◽  
Integrable Systems ◽  
Minimal Surfaces ◽  
Gauss Map ◽  
Surface Theory ◽  
Recent Developments ◽  
Hopf Differential ◽  
Simple Factor ◽  
Costa Surface

AbstractIn this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez–Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well–known examples of minimal surfaces, namely the Riemann minimal surfaces and the Costa surface.

scholarly journals On the Gauss map of minimal surfaces with finite total curvature

1991 ◽  
Vol 44 (2) ◽  
pp. 225-232 ◽  
Author(s):  
Min Ru
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
General Position ◽  
Total Curvature ◽  
Gauss Map ◽  
Finite Total Curvature

We prove that if a nonflat complete regular minimal surface immersed in Rn is of finite total curvature, then its Gauss map can omit at most (n – 1)(n + 2)/2 hyperplanes in general position in Pn–1 (ℂ).

How many maximal surfaces do correspond to one minimal surface?

2009 ◽  
Vol 146 (1) ◽  
pp. 165-175 ◽  
Author(s):  
HENRIQUE ARAÜJO ◽  
MARIA LUIZA LEITE
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Hausdorff Space ◽  
Quotient Space ◽  
Closed Interval ◽  
Congruence Class ◽  
Quotient Topology ◽  
Maximal Surfaces ◽  
Hopf Differential ◽  
Congruence Classes

AbstractWe discuss the minimal-to-maximal correspondence between surfaces and show that, under this correspondence, a congruence class of minimal surfaces in 3 determines an 2-family of congruence classes of maximal surfaces in 3. It is proved that further identifications among these classes may exist, depending upon the subgroup of automorphisms preserving the Hopf differential on the underlying Riemann surface. The space of maximal congruence classes inherits a quotient topology from 2. In the case of the Scherk minimal surface, the subgroup has order two and the quotient space is topologically a disc with boundary. Other classical examples are discussed: for the Enneper minimal surface, one obtains a non-Hausdorff space; for the minimal catenoid, a closed interval.

scholarly journals The Gauss Map of Complete Minimal Surfaces with Finite Total Curvature

2013 ◽  
Vol 85 (4) ◽  
pp. 1217-1226
Author(s):  
PEDRO A. HINOJOSA ◽  
GILVANEIDE N. SILVA
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Total Curvature ◽  
Gauss Map ◽  
Additional Hypothesis ◽  
Finite Total Curvature

In this paper we are concerned with the image of the normal Gauss map of a minimal surface immersed in ℝ3 with finite total curvature. We give a different proof of the following theorem of R. Osserman: The normal Gauss map of a minimal surface immersed in ℝ3 with finite total curvature, which is not a plane, omits at most three points of��2 Moreover, under an additional hypothesis on the type of ends, we prove that this number is exactly 2.

scholarly journals NEW COMPLEX ANALYTIC METHODS IN THE THEORY OF MINIMAL SURFACES: A SURVEY

2018 ◽  
Vol 106 (03) ◽  
pp. 287-341 ◽  
Author(s):  
ANTONIO ALARCÓN ◽  
FRANC FORSTNERIČ
Keyword(s):  
Minimal Surfaces ◽  
Gauss Map ◽  
Holomorphic Maps ◽  
Convex Domains ◽  
Global Theory ◽  
Open Riemann Surface ◽  
Analytic Methods ◽  
Recent Developments ◽  
Minimal Immersions ◽  
Hilbert Boundary Value Problem

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann–Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi–Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^{n}$ and in minimally convex domains of $\mathbb{R}^{n}$ , results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

scholarly journals Minimal surfaces for architectural constructions

2008 ◽  
Vol 6 (1) ◽  
pp. 89-96 ◽  
Author(s):  
Ljubica Velimirovic ◽  
Grozdana Radivojevic ◽  
Mica Stankovic ◽  
Dragan Kostic
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Recent Time ◽  
Surface Theory ◽  
The Family ◽  
Minimal Area

Minimal surfaces are the surfaces of the smallest area spanned by a given boundary. The equivalent is the definition that it is the surface of vanishing mean curvature. Minimal surface theory is rapidly developed at recent time. Many new examples are constructed and old altered. Minimal area property makes this surface suitable for application in architecture. The main reasons for application are: weight and amount of material are reduced on minimum. Famous architects like Otto Frei created this new trend in architecture. In recent years it becomes possible to enlarge the family of minimal surfaces by constructing new surfaces.

ON ALGEBRAIC MINIMAL SURFACES IN ℝ3 DERIVING FROM CHARGE 2 MONOPOLE SPECTRAL CURVES

2005 ◽  
Vol 16 (02) ◽  
pp. 173-180 ◽  
Author(s):  
ANTHONY SMALL
Keyword(s):  
Line Bundle ◽  
Minimal Surface ◽  
Elliptic Curves ◽  
Gaussian Curvature ◽  
Minimal Surfaces ◽  
Gauss Map ◽  
Spectral Curves

We give formulae for minimal surfaces in ℝ3 deriving, via classical osculation duality, from elliptic curves in a line bundle over ℙ1. Specialising to the case of charge 2 monopole spectral curves we find that the distribution of Gaussian curvature on the auxiliary minimal surface reflects the monopole's structure. This is elucidated by the behaviour of the surface's Gauss map.

scholarly journals V.M. Miklyukov: from Dimension 8 to Nonassociative Algebras

2019 ◽  
pp. 33-50 ◽  
Author(s):  
Vladimir Tkachev
Keyword(s):  
Minimal Surface ◽  
Surface Theory ◽  
Short Survey ◽  
Nonassociative Algebras ◽  
Recent Developments ◽  
Minimal Cones

In this short survey we give a background and explain some recent developments in algebraic minimal cones and nonassociative algebras. A part of this paper is recollections of my collaboration with my teacher, PhD supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that motivated the present research.

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.

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