Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space

Yıl 2021, Cilt: 9 Sayı: 1, 132 - 136, 28.04.2021

Erhan Güler , Ömer Kişi

Öz

We consider a two parameter family of Henneberg-type minimal surfaces $ \mathfrak{H}_{m,n}$ using the Weierstrass representation in the four dimensional Euclidean space $\mathbb{E}^{4}$. An invariant linear map of Weingarten type in the tangent space of the Henneberg-type minimal surface $\mathfrak{H}_{4,2}$ which generates two invariants $\kappa $ and $\varkappa $, is characterized by $ \varkappa^{2}=\kappa $ in $\mathbb{E}^{4}$.

Anahtar Kelimeler

Four dimension, Henneberg-type minimal surface, Weierstrass representation, Weingarten map

Kaynakça

• [1] K. Arslan, B. Bulca, V. Milousheva, Meridian Surfaces in E4. with Pointwise 1-type Gauss Map, Bull. Korean Math. Soc. 51 (2014) 911–922.
• [2] K. Arslan, R. Deszcz, S. Yaprak, On Weyl Pseudosymmetric Hypersurfaces, Colloq. Math. 72(2) (1997) 353 361.
• [3] B.Y. Chen, Geometry of Submanifolds. Pure and Applied Mathematics, No. 22. Marcel Dekker, Inc., New York, 1973.
• [4] U. Dierkes, S. Hildebrandt, F. Sauvigny, Minimal Surfaces, Springer-Verlag, Berlin, Heidelberg. 2nd ed. 2010.
• [5] L.P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications, N.Y. 1909.
• [6] G. Ganchev, V. Milousheva, On the Theory of Surfaces in the Four-Dimensional Euclidean Space, Kodai Math. J. 31 (2008) 183-198.
• [7] G. Ganchev, V. Milousheva, An Invariant Theory of Surfaces in the Four-Dimensional Euclidean or Minkowski Space, Pliska Stud. Math. Bulgar. 21 (2012) 177 200.
• [8] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., CRC Press, Boca Raton, FL. 1998.
• [9] E. Guler, H.H. Hacısalihoglu, Y.H. Kim, The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space, Symmetry 10(9) (2018) 1-11.
• [10] E. Guler, O. Kisi, Weierstrass Representation, Degree and Classes of the Surfaces in the Four Dimensional Euclidean Space, Celal Bayar Un. J. Sci., 13-1 (2017) 155-163.
• [11] E. Guler, O. Kisi, C. Konaxis, Implicit Equation of the Henneberg-Type Minimal Surface in the Four Dimensional Euclidean Space, Mathematics Sp. Iss. : Comp. Alg. Sci. Comp. 6(12) (2018) 1 10.
• [12] E. Guler, M. Magid, Y. Yaylı, Laplace Beltrami Operator of a Helicoidal Hypersurface in Four Space, J. Geom. Sym. Phys. 41 (2016) 77–95.
• [13] E. Guler, N.C. Turgay, Cheng-Yau operator and Gauss map of rotational hypersurfaces in 4-space, Mediterr. J. Math. 16(3) (2019) 1–16.
• [14] L. Henneberg, Uber Salche Minimalflache, Welche Eine Vorgeschriebene Ebene Curve Sur Geodatishen¨ Line Haben, Doctoral Dissertation, Eid- genossisches Polythechikum, Zurich 1875.
• [15] L. Henneberg, Uber Diejenige Minimalflache,¨ Welche Die Neil’sche Paralee Zur Ebenen Geodatischen¨ Line Hat, Vierteljschr Natuforsch, Ges. Zurich,¨ 21 (1876) 66 70.
• [16] L. Henneberg, Bestimmung Der Neidrigsten Classenzahl Der Algebraischen Minimalflachen¨, Annali di Matem. Pura Appl. 9 (1878) 54 57.
• [17] D.A. Hoffman, R. Osserman, The Geometry of the Generalized Gauss Map, Memoirs of the AMS, 1980.
• [18] F. Kahraman Aksoyak, Y. Yaylı, Boost Invariant Surfaces with Pointwise 1-type Gauss Map in Minkowski 4-Space E41; Bull.,Korean Math. Soc. 51 (2014) 1863 1874.
• [19] F. Kahraman Aksoyak, Y. Yaylı, General Rotational surfaces with Pointwise 1-type Gauss Map in Pseudo-Euclidean Space E42, Indian J. Pure Appl. Math. 46 (2015) 107–118.
• [20] C. Moore, Surfaces of Rotation in a Space of Four Dimensions, The Annals of Math., 2nd Ser., 21(2) (1919) 81 93.
• [21] J.C.C. Nitsche, Lectures on Minimal Surfaces. Vol. 1. Introduction, fundamentals, geometry and basic boundary value problems, Cambridge Un Press, Cambridge. 1989.
• [22] M.E.G.G. de Oliveira, Some New Examples of Nonorientable Minimal Surfaces, Proc. Amer. Math. Soc. 98-4 (1986) 629 636.
• [23] R. Osserman, A Survey of Minimal Surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne. 1969.
• [24] K. Weierstrass, Untersuchungen Uber Die Flachen, Deren Mittlere Krummung¨ Uberall Gleich Null Ist., Monatsber d Berliner Akad. (1866) 612 625.
• [25] K. Weierstrass, Mathematische Werke, Vol. 3, Mayer & Muller, Berlin, 1903.