The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space

The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space

We consider the Enneper family of real maximal surfaces via Weierstrass data <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>,</mo><msup>&l...

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Main Author: Erhan Güler
Format: Article
Language:English
Published: MDPI AG 2021-12-01
Series:Axioms
Subjects:
Weierstrass representation
Enneper maximal surface
algebraic surface
degree
class
Online Access:https://www.mdpi.com/2075-1680/11/1/4
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author Erhan Güler
author_facet Erhan Güler
author_sort Erhan Güler
collection DOAJ
description We consider the Enneper family of real maximal surfaces via Weierstrass data <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>,</mo><msup><mi>ζ</mi><mi>m</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>∈</mo><msub><mi mathvariant="double-struck">Z</mi><mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">E</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula>. Moreover, we propose that the family has degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></semantics></math></inline-formula> (resp., class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>m</mi><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>) in the cartesian coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></semantics></math></inline-formula> (resp., in the inhomogeneous tangential coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></semantics></math></inline-formula>).
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spelling doaj.art-0e608a2fae6c41d3ba0c74a1dff6200b2023-11-23T12:58:02ZengMDPI AGAxioms2075-16802021-12-01111410.3390/axioms11010004The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski SpaceErhan Güler0Department of Mathematics, Faculty of Sciences, Kutlubey Campus, Bartın University, Bartın 74100, TurkeyWe consider the Enneper family of real maximal surfaces via Weierstrass data <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>,</mo><msup><mi>ζ</mi><mi>m</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>∈</mo><msub><mi mathvariant="double-struck">Z</mi><mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">E</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula>. Moreover, we propose that the family has degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></semantics></math></inline-formula> (resp., class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>m</mi><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>) in the cartesian coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></semantics></math></inline-formula> (resp., in the inhomogeneous tangential coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></semantics></math></inline-formula>).https://www.mdpi.com/2075-1680/11/1/4Weierstrass representationEnneper maximal surfacealgebraic surfacedegreeclass
spellingShingle Erhan Güler
The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space
Axioms
Weierstrass representation
Enneper maximal surface
algebraic surface
degree
class
title The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space
title_full The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space
title_fullStr The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space
title_full_unstemmed The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space
title_short The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space
title_sort algebraic surfaces of the enneper family of maximal surfaces in three dimensional minkowski space
topic Weierstrass representation
Enneper maximal surface
algebraic surface
degree
class
url https://www.mdpi.com/2075-1680/11/1/4
work_keys_str_mv AT erhanguler thealgebraicsurfacesoftheenneperfamilyofmaximalsurfacesinthreedimensionalminkowskispace
AT erhanguler algebraicsurfacesoftheenneperfamilyofmaximalsurfacesinthreedimensionalminkowskispace

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