POLİTEKNİK DERGİSİ
JOURNAL of POLYTECHNIC
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Monge hypersurfaces in euclidean 4-space with density Yoğunluklu öklidyen 4-uzayında monge hiperyüzeyleri
Yazar(lar) (Author(s)): Mustafa ALTIN1, Ahmet KAZAN2, H. Bayram KARADAĞ3
ORCID1: 0000-0001-5544-5910 ORCID2: 0000-0002-1959-6102 ORCID3: 0000-0001-6474-877X
Bu makaleye şu şekilde atıfta bulunabilirsiniz(To cite to this article): Altın M., Kazan A. ve Karadağ H. B., “Monge hypersurfaces in euclidean 4-space with density”, Politeknik Dergisi, 23(1): 207-214, (2020).
Erişim linki (To link to this article): http://dergipark.org.tr/politeknik/archive
DOI: 10.2339/politeknik.634175 Politeknik Dergisi, 2020; 23(1) : 207-214 Journal of Polytechnic, 2020; 23 (1): 207-214
Yoğunluklu Öklidyen 4-Uzayında Monge Hiperyüzeyleri Araştırma Makalesi / Research Article Mustafa ALTIN1, Ahmet KAZAN2*, H. Bayram KARADAĞ3 1Technical Sciences Vocational School, Bingol University, Bingol, Turkey 2Doğanşehir Vahap Küçük Vocational School of Higher Education, Department of Computer Technologies, Malatya Turgut Özal University, Turkey 3Faculty of Arts and Sciences, Department of Mathematics, Inonu University, Turkey (Geliş/Received : 17.10.2019 ; Kabul/Accepted : 31.10.2019)
ÖZ Bu çalışmada, ilk olarak 4-boyutlu Öklidyen uzayında bir Monge hiperyüzeyinin ortalama ve Gaussian eğriliklerini verdik. Ardından, farklı yoğunluklara sahip uzayında Monge hiperyüzeylerini çalıştık. Bu bağlamda, ve hepsi aynı anda sıfır olmayan sabitler olmak üzere, (lineer yoğunluk) ve yoğunluklu uzayında ağırlıklı minimal ve ağırlıklı flat Monge hiperyüzeylerini ve sabitlerinin farklı seçimleri yardımıyla elde ettik. Anahtar Kelimeler:Yoğunluklu manifold, ağırlıklı ortalama eğrilik, ağırlıklı gaussian eğriliği, monge yüzeyleri. Monge Hypersurfaces in Euclidean 4-Space with Density
ABSTRACT In the present study, firstly we give the mean and Gaussian curvatures of a Monge hypersurface in 4-dimensional Euclidean space. After this, we study on Monge hypersurfaces in with different densities. In this context, we obtain the weighted minimal and weighted flat Monge hypersurfaces in with densities (linear density) and with the aid of different choices of constants and , where and are not all zero constants. Keywords: Manifold with density, weighted mean curvature, weighted gaussian curvature, monge hypersurfaces 1. INTRODUCTION curvature vanishes in [8]. Yoon has investigated the Minimal and flat surfaces have long been an important rotational surfaces with finite type Gauss map in topic of study by mathematicians and other scientists. Euclidean 4-space. He has proved that, the Gauss map is When we focus on the studies on this subject, some of of finite type if and only if rotatinal surface is a Clifford these studies can be given as follows: In the first two torus [4]. Dursun and Turgay have studied general decades of 1900s, Moore has studied rotational surfaces rotational surfaces in 퐸4 whose meridian curves lie in and rotational surfaces with constant curvature in four- two-dimensional planes and they have found all minimal dimensional space and he has given some relations for general rotational surfaces by solving the differential them, [1,2]. Moor’s studies have examined by Ganchev equation that characterizes minimal general rotational and Milousheva in Minkowski 4-space and some surfaces. Also, they have determined all pseudo- relations have been expressed, [7]. In [3], complete umbilical general rotational surfaces in 퐸4, [9]. 4 hypersurfaces in ℝ with constant mean curvature and Kahraman and Yaylı have studied Bost invariant surfaces constant scalar curvature have been classified. In [5,6], 4 with pointwise 1-type Gauss map in 퐸1 and they have authors have studied generalized rotational surfaces and generalized rotational surfaces of pointwise 1-type Gauss translation surfaces in 4-dimensional Euclidean surfaces 4 map in 퐸2 [10,11]. Güler and et al have defined helicoidal and they have investigated curvature properties of these hypersurface with the Laplace-Beltrami operator in four surfaces and they have given some examples for them. space, [12]. Also, Güler and et al have studied Gauss map Also authors have proved that, a translation surface is flat and the third Laplace-Beltrami operator of the rotational if and only if it is a hyperplane or a hypercylinder. Moruz hypersurface in 4-space, [13]. Since, the curvature of a 4 and Mounteanu have considered hypersurfaces in ℝ curve and the mean curvature of an n-dimensional defined as the sum of a curve and a surface whose mean hypersurface are important invariants for curves and *Sorumlu Yazar (Corresponding Author) surfaces, many authors have studied these notions for e-posta : [email protected] different types of curves and surfaces for a long time in
207 Mustafa ALTIN, Ahmet KAZAN, H. Bayram KARADAĞ / POLİTEKNİK DERGİSİ, Politeknik Dergisi,2020;23(1): 207-214 different spaces, such as Euclidean, Minkowski, Galilean Also, Corvin and et al have introduced the notion of and pseudo-Galilean spaces. generalized weighted Gaussian curvature on a manifold Now, let us recall some fundamental notions in Euclidean as 4-space. 퐺휙 = 퐺 −△ 휑 , (1.11)
Let 푥⃗ = (푥1, 푦1,푧1, 푡1), 푦⃗ = (푥2, 푦2, 푧2, 푡2) and 푧⃗ = where △ is the Laplacian operator [15]. A hypersurface 4 (푥3, 푦3,푧3, 푡3) be three vectors in 퐸 . Then, the inner is called weighted flat (or 휑-flat), if its weighted product and vector product of these vectors are given by Gaussian curvature vanishes.
⟨푥⃗, 푦⃗⟩ = 푥1푥2 + 푦1푦2 + 푧1푧2 + 푡1푡2 (1.1) After these definitions, lots of studies have been done by and differential geometers about weighted manifolds, for instance [16-25]. 푒1 푒2 푒3 푒4 푥 푦 푧 푡 푥⃗ × 푦⃗ × 푧⃗ = 푑푒푡 ( 1 1 1 1 ), (1.2) 푥2 푦2 푧2 푡2 2. MONGE HYPERSURFACES IN EUCLIDEAN 푥3 푦3 푧3 푡3 4-SPACE respectively. If In this section, we obtain the Gaussian and mean 3 4 푋: 퐸 ⟶ 퐸 , (푢1, 푢2, 푢3) ⟶ 푋(푢1, 푢2, 푢3) (1.3) curvatures of a Monge hypersurface in Euclidean 4- space, by giving the normal vector field of it. = (푋1(푢1, 푢2, 푢3), 푋2(푢1, 푢2, 푢3), 푋3(푢1,푢2, 푢3), 푋4(푢1, 푢2, 푢3)) 4 is a hypersurface in Euclidean 4-space 퐸4, then the Let 푀 be a surface in 퐸 given by normal vector field, the matrix forms of the first and 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푓(푥, 푦, 푧)). (2.1) second fundamental forms are Then we call this surface as Monge hypersurface in 푋 ×푋 ×푋 푁 = 푢1 푢2 푢3 , (1.4) Euclidean 4-space. For this surface, we have ‖푋푢1×푋푢2×푋푢3‖ 푋 = (1,0,0, 푓 ), 푋 = (0,1,0,푓 ), 푋 = (0,0,1, 푓 ) 푔11 푔12 푔13 푥 푥 푦 푦 푧 푧 푔 푔 푔 푔푖푗 = [ 21 22 23] (1.5) 푋푥푥 = (0,0,0,푓푥푥), 푋푥푦 = (0,0,0,푓푥푦), (2.2) 푔 푔 푔 31 32 33 푋 = (0,0,0,푓 ), 푋 = (0,0,0,푓 ), and 푥푧 푥푧 푦푦 푦푦 푋푦푧 = (0,0,0,푓푦푧), 푋푧푧 = (0,0,0, 푓푧푧), ℎ11 ℎ12 ℎ13 ℎ = [ℎ ℎ ℎ ], (1.6) 휕푋 휕푋 휕푓 휕푓 푖푗 21 22 23 where 푋푖 = , 푋푖푗 = , 푓푖 = , 푓푖푗 = , {푖, 푗} ∈ ℎ ℎ ℎ 휕푖 휕푖푗 휕푖 휕푖푗 31 32 33 {푥, 푦}. From (2.2), respectively. Here, 푔 = 〈푋 , 푋 〉, ℎ = 〈푋 , 푁〉, 푖푗 푢푖 푢푗 푖푗 푢푖푢푗 푒1 푒2 푒3 푒4 휕푋 휕2푋 1 0 0 푓푥 푋푢 = , 푋푢 푢푗 = , {푖, 푗} ∈ {1,2,3}. 푖 휕푢 푖 휕푢 푢 푋푥 × 푋푦 × 푋푧 = | | = (푓푥, 푓푦, 푓푧, −1) (2.3) 푖 푖 푗 0 1 0 푓푦 Also, the shape operator of the hypersurface (1.3) is 0 0 1 푓푧 −1 and so, from (1.2) and (1.4) the normal vector field of the 푆 = (푎푖푗) = (ℎ푖푗).(푔푖푗) , (1.7) −1 surface (2.1) is obtained as where (푔푖푗) is the inverse matrix of (푔푖푗). (푓 ,푓 ,푓 ,−1) 푁 = 푥 푦 푧 . (2.4) 2 2 2 With the aid of (1.5)-(1.7), the Gaussian curvature and √1+푓푥 +푓푦 +푓푧 mean curvature of a hypersurface in 퐸4 are given by Also from (1.6), the matrix form of the second 푑푒푡(ℎ ) 퐾 = 푖푗 (1.8) fundamental form of the surface (2.1) is 푑푒푡(푔푖푗) −푓푥푥 −푓푥푦 −푓푥푧 and 1 (ℎ푖푗) = [−푓푥푦 −푓푦푦 −푓푦푧] (2.5) 2 2 2 3퐻 = 푖푧(푆), (1.9) √1+푓푥 +푓푦 +푓푧 −푓푥푧 −푓푦푧 −푓푧푧 respectively. and its determinant is Furthermore, the notion of weighted manifold which is −푓 푓 푓 −2푓 푓 푓 +푓 푓2 +푓 푓2 +푓 푓2 푑푒푡(ℎ푖푗) = 푥푥 푦푦 푧푧 푥푦 푦푧 푥푧 푥푥 푦푧 푦푦 푥푧 푧푧 푥푦. (2.6) an important topic for geometers and physicists has been 2 2 2 3/2 studied by many scientists, recently. Firstly, Gromov has (1+푓푥 +푓푦 +푓푧 ) introduced the notion of weighted mean curvature (or 휑- Now, we obtain the matrix of the metric 푔푖푗, its mean curvature) of an n-dimensional hypersurface as determinant and inverse as 1 푑휑 2 1 + 푓푥 푓푥푓푦 푓푥푓푧 퐻휑 = 퐻 − , (1.10) (푛−1) 푑푁 2 푔푖푗 = [푓푥푓푦 1 + 푓푦 푓푦푓푧 ], (2.7) where 퐻 is the mean curvature and 푁 is the unit normal 2 vector field of the surface [14]. A hypersurface is called 푓푥푓푧 푓푧푓푦 1 + 푓푧 weighted minimal (or 휑-minimal), if its weighted mean 2 2 2 2 푑푒푡(푔푖푗) = (1 + 푓푥 ) [(1 + 푓푦 )(1 + 푓푧 ) − (푓푦푓푧) ] curvature vanishes. 2 2 − 푓푥푓푦[푓푥푓푦(1 + 푓푧 ) − 푓푥푓푦푓푧 ]
208 MONGE HYPERSURFACES IN EUCLIDEAN 4-SPACE WITH DENSITY- … Politeknik Dergisi, 2020; 23 (1) : 207-214
2 2 2 +푓푥푓푧[푓푥푓푧푓푦 − (1 + 푓푦 )푓푥푓푧] where ℎ, 푔 and 푚 are 퐶 −differentiable functions. Thus, 2 2 2 we have = 1 + 푓푥 + 푓푦 + 푓푧 (2.8) 푓 = ℎ′(푥), 푓 = 푔′(푦), 푓 = 푚′(푧), and 푥 푦 푧 ′′ 2 2 푓 = ℎ (푥), 푓 = 0, 푓 = 0, (3.3) 1 + 푓푦 + 푓푧 −푓푥푓푦 −푓푥푓푧 푥푥 푥푦 푥푧 −1 1 2 2 ′′ ′′ (푔푖푗) = 2 2 2 [−푓푥푓푦 1 + 푓푥 + 푓푧 −푓푦푓푧 ], (2.9) ( ) ( ) 1+푓푥 +푓푦 +푓푧 푓푦푦 = 푔 푦 , 푓푦푧 = 0, 푓푧푧 = 푚 푧 . −푓 푓 −푓 푓 1 + 푓2 + 푓2 푥 푧 푧 푦 푥 푦 Using (3.3) in (3.1), the weighted mean curvature of the respectively. Hence, using (2.6) and (2.8) in (1.8), we surface (2.1) is obtained as obtain the Gaussian curvature of the surface (2.1) as −ℎ′′(푥)(1+푔′(푦)2+푚′(푧)2) 2 2 ( )2 −푓푥푥푓푧푧푓푦푦−2(푓푥푦푓푦푧푓푥푧)+푓푥푥(푓푦푧) +푓푦푦 푓푥푧 +푓푧푧(푓푥푦) −푔′′(푦)(1+ℎ′(푥)2+푚′(푧)2) 퐾 = 5/ . (2.10) 2 2 2 2 ′′ ′ 2 ′ 2 (1+푓푥 +푓푦 +푓푧 ) −푚 (푧)(1+ℎ (푥) +푔 (푦) )− ′ ′ ′ ′ 2 ′ 2 ′ 2 −1 {(훼ℎ (푥)+훽푔 (푦)+훾푚 (푧)−휇)(1+ℎ (푥) +푔 (푦) +푚 (푧) )} Let we take (푎푖푗) = (ℎ푖푗) × (푔푖푗) . Then, since 퐻 = . (3.4) 휑 3(1+ℎ′(푥)2+푔′(푦)2+푚′(푧)2)3/2 (훼푖푗 ) = (2.11) 2 2 −푓푥푥 −푓푥푦 −푓푥푧 1 + 푓푥 + 푓푧 −푓푥푓푦 −푓푥푓푧 1 2 2 Proposition 2. Let 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + [−푓푥푦 −푓푦푦 −푓푦푧] . [−푓푥푓푦 1 + 푓푥 + 푓푧 −푓푦푓푧 ] 2 2 2 3/2 (1 + 푓푥 + 푓푦 + 푓푧 ) −푓 −푓 −푓 2 2 푥푧 푦푧 푧푧 −푓푥푓푧 −푓푧푓푦 1 + 푓푥 + 푓푦 푔(푦) + 푚(푧)) be a Monge hypersurface in Euclidean 4- from (1.9), we obtain the mean curvature of the surface space with linear density 푒훼푥+훽푦+훾푧+휇푡, where 훼, 훽, 훾 (2.1) as and 휇 are not all zero constants. Then, this surface is 2 2 2 2 2 2 −푓푥푥(1+푓푦 +푓푧 )−푓푦푦(1+푓푥 +푓푧 )−푓푧푧(1+푓푥 +푓푦 ) weighted minimal if and only if { } +2(푓 푓 푓 +푓 푓 푓 +푓 푓 푓 ) ′′ ′ 2 ′ 2 퐻 = 푥푦 푥 푦 푥푧 푥 푧 푦푧 푦 푧 . (2.12) 0 = ℎ (푥)(1 + 푔 (푦) + 푚 (푧) ) + 2 2 3/2 ′′ ′ 2 ′ 2 3(1+푓푥 +푓푦 +푓푧) 푔 (푦)(1 + ℎ (푥) + 푚 (푧) ) + (3.5) 푚′′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2) + 3. MONGE HYPERSURFACES IN 퐄ퟒ WITH (훼ℎ′(푥) + 훽푔′(푦) + 훾푚′(푧) − 휇)(1 + ℎ′(푥)2 + 푔′(푦)2 + LINEAR DENSITY 푚′(푧)2) (3.5) In the first subsection of this section, we investigate the satisfies. weighted minimal Monge hypersurfaces in Euclidean 4- Next, we’ll obtain the weighted minimal Monge space with linear density 푒훼푥+훽푦+훾푧+휇푡 and in the second hypersurfaces in 퐸4 with density 푒훼푥+훽푦+훾푧+휇푡 for subsection of this section, we investigate the weighted different choices of the not all zero constants 훼, 훽, 훾 and flat Monge hypersurfaces in 퐸4 with this density. 휇. ퟒ 3.1. Weighted Minimal Monge Hypersurfaces in 푬 We note that, throughout this study we consider 푘푖 and + with Linear Density 휆푖, 푖 ∈ ℕ , are real constants. Let 푀 be a Monge hypersurface given by (2.1) in Case 1. Let the density be 푒훼푥: 푎푥+훽푦+훾푧+휇푡 Euclidean 4-space with linear density 푒 , In this case, let us consider the Monge hypersurface where 훼, 훽, 훾 and 휇 are not all zero constants. Then from ( ) ( ) ( ) ( ) (1.10), the weighted mean curvature of this surface is 푀: 푋 푥, 푦, 푧 = (푥, 푦, 푧, ℎ 푥 + 푔 푦 + 푚 푧 ) obtained as in Euclidean 4-space with linear density 푒훼푥. Then, this −푓 (1+푓2+푓2)−푓 (1+푓2+푓2) surface is weighted minimal if and only if 푥푥 푦 푧 푦푦 푥 푧 2 2 ′′ ′ 2 ′ 2 −푓푧푧(1+푓푥 +푓푦 )+ 0 = ℎ (푥)(1 + 푔 (푦) + 푚 (푧) ) + 2(푓푥푦푓푥푓푦+푓푥푧푓푥푓푧+푓푦푧푓푦푓푧)− ′′ ′ 2 ′ 2 푔 (푦)(1 + ℎ (푥) + 푚 (푧) )+ (3.6) {(훼푓 +훽푓 +훾푓 −휇)(1+푓2+푓2+푓2)} 퐻 = 푥 푦 푧 푥 푦 푧 . (3.1) ′′ ′ 2 ′ 2 휑 2 2 2 3/2 푚 (푧)(1 + ℎ (푥) + 푔 (푦) )+ 3(1+푓푥 +푓푦 +푓푧 ) 훼ℎ′(푥)(1 + ℎ′(푥)2 + 푔′(푦)2 + 푚′(푧)2) So, we have satisfies. Here, by obtaining some special solutions for ( ) ( ) Proposition 1. Let 푀: 푋 푥, 푦, 푧 = (푥, 푦, 푧, 푓 푥, 푦, 푧 ) be the equation (3.6), we’ll construct the weighted minimal a Monge hypersurface in Euclidean 4-space with linear Monge hypersurfaces in 퐸4 with linear density 푒훼푥. density 푒훼푥+훽푦+훾푧+휇푡, where 훼, 훽, 훾 and 휇 are not all Firstly, let us take the functions ( ) and ( ) are linear, zero constants. Then, this surface is weighted minimal if 푔 푦 푚 푧 ( ) ( ) and only if i.e. 푔 푦 = 푘1푦 + 푘2, 푚 푧 = 푘3푧 + 푘4. Then, the equation (3.6) becomes 2(푓푥푦푓푥푓푦 + 푓푥푧푓푥푓푧 + 푓푦푧푓푦푓푧)= ′′ 2 2 2 2 2 2 2 2 ℎ (푥)(1 + (푘1) + (푘3) ) = 푓푥푥(1 + 푓푦 + 푓푧 ) + 푓푦푦(1 + 푓푥 + 푓푧 ) + 푓푧푧(1 + 푓푥 + 푓푦 ) ′ ′ 2 2 2 2 2 2 −훼ℎ (푥)(1 + (ℎ ) + (푘1) + (푘3) ). (3.7) +(훼푓푥 + 훽푓푦 + 훾푓푧 − 휇)(1 + 푓푥 + 푓푦 + 푓푧 ) (3.2) From (3.7), satisfies. ℎ′′(1 + (푘 )2 + (푘 )2) Now, let we take 1 3 = −1 훼ℎ′(1 + (ℎ′)2 + (푘 )2 + (푘 )2) 푓(푥, 푦, 푧) = ℎ(푥) + 푔(푦) + 푚(푧), 1 3
209 Mustafa ALTIN, Ahmet KAZAN, H. Bayram KARADAĞ / POLİTEKNİK DERGİSİ, Politeknik Dergisi,2020;23(1): 207-214
′′ ′′ ′ 훼푘5푦 2 2 ℎ ℎ ℎ 푙푛(푐표푠( +휆3))(1+(푘5) +(푘3) ) √1+(푘 )2+(푘 )2 ⟹ ′ − ′ 2 2 2 = −훼 5 3 ℎ (ℎ ) + 1 + (푘1) + (푘3) ⟹ 푔(푦) = + 휆4. (3.11) 훼푘5 ′ ′ 1 ′ 2 2 2 Hence, we have ⟹ (푙푛|ℎ | − 푙푛|(ℎ ) + 1 + (푘1) + (푘3) |) = −훼 2 Theorem 2. The weighted minimal Monge hypersurface ℎ′ in Euclidean 4-space with linear density 푒훼푥 for (훼 ≠ ⟹ 푙푛 | | = −훼푥 + 휆1 ′ 2 2 2 0) ∈ ℝ can be parametrized by √(ℎ ) + 1 + (푘1) + (푘3) ℎ′ 푋(푥, 푦, 푧) =( 푥, 푦, 푧, 푘5푥 + 푘3푧 + 푘 −훼푥+휆1 ⟹ = 푒 훼푘5푦 2 2 ′ 2 2 2 푙푛 (푐표푠 ( +휆3))(1+(푘5) +(푘3) ) √(ℎ ) + 1 + (푘1) + (푘3) 2 2 √1+(푘5) +(푘3) ′ −훼푥+휆1 ′ 2 2 2 + ) , (3.12) ⟹ ℎ = 푒 √(ℎ ) + 1 + (푘1) + (푘3) 훼푘5 ′ 2 −2훼푥+2휆1 ′ 2 2 2 where 푘 = 푘 +푘 + 휆 . ⟹ (ℎ ) = 푒 ((ℎ ) + 1 + (푘1) + (푘3) ) 4 6 4 −2훼푥+2휆1 ′ 2 −2훼푥+2휆1 2 2 And now, taking the functions ℎ(푥) and 푔(푦) are linear, ⟹ (1 − 푒 )(ℎ ) = 푒 (1 + (푘1) + (푘3) ) i.e. ℎ(푥) = 푘5푥 + 푘6, 푔(푦) = 푘1푦 + 푘2, from (3.6), we √1 + (푘 )2 + (푘 )2. 푒−훼푥+휆1 ⟹ ℎ′ = 1 3 have √1 − (푒−훼푥+휆1)2 푚′(푧)2 푚′′(푧) = −훼푘 (1 + ). (3.13) 5 1+(푘 )2+(푘 )2 √(1 + (푘 )2 + (푘 )2)푎푟푐푡푎푛(푒−휆1 √푒2훼푥 − 푒2휆1 ) 5 1 ⟹ ℎ(푥) = 1 3 + 휆 . 훼 2 Solving (3.13) with the same procedure as above, we (3.8) have 훼푘5푧 2 2 푙푛(푐표푠( +휆5))(1+(푘5) +(푘1) ) Thus, 2 2 √1+(푘5) +(푘1) 2 2 −휆 2훼푥 2휆 ( ) . √(1+(푘1) +(푘3) )푎푟푐푡푎푛(푒 1√푒 −푒 1) 푚 푧 = + 휆6 (3.14) 푓(푥, 푦, 푧) = 훼푘5 훼 So, we get +푘1푦 + 푘3푧 + 푘2 + 푘4 + 휆2 Theorem 3. The weighted minimal Monge hypersurface is a solution of (3.7). in Euclidean 4-space with linear density 푒훼푥 for (훼 ≠ So, we can give the following Theorem: 0) ∈ ℝ can be parametrized by
Theorem 1. The weighted minimal Monge hypersurface 푋(푥, 푦, 푧) =( 푥, 푦, 푧, 푘5푥 + 푘1푦 + 푘 + 훼푥 in Euclidean 4-space with linear density 푒 for (훼 ≠ 훼푘5푧 2 2 푙푛(푐표푠( +휆5))(1+(푘5) +(푘1) ) 2 2 0) ∈ ℝ can be parametrized by √1+(푘5) +(푘1) ), (3.15) 훼푘 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푘1푦 + 푘3푧 + 푘 5 2 2 −휆 2훼푥 2휆 where 푘 = 푘 + 푘 + 휆 . √(1+(푘1) +(푘3) )푎푟푐푡푎푛(푒 1√푒 −푒 1) 6 2 6 ), (3.9) 훽푦 훼 Case 2. Let the density be 푒 : where 푘 = 푘2 + 푘4 + 휆2. In this case, let us consider the Monge hypersurface Secondly, taking the functions ℎ(푥) and 푚(푧) are linear, 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 푚(푧)) i.e. ℎ(푥) = 푘5푥 + 푘6, 푚(푧) = 푘3푧 + 푘4, from (3.6), we 훽푦 have in Euclidean 4-space with linear density 푒 . Then, this surface is weighted minimal if and only if 푔′′(푦)(1 + (푘 )2 + (푘 )2) = 5 3 0 = ℎ′′(푥)(1 + 푔′(푦)2 + 푚′(푧)2) −훼푘 (1 + (푔′(푦))2 + (푘 )2 + (푘 )2). (3.10) 5 5 3 +푔′′(푦)(1 + ℎ′(푥)2 + 푚′(푧)2) (3.16) Solving this equation, we reach that +푚′′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2) (푔′)2 푔′′ = −훼푘 (1 + ) +훽푔′(푦)(1 + ℎ′(푥)2 + 푔′(푦)2 + 푚′(푧)2) 5 1 + (푘 )2 + (푘 )2 5 3 satisfies. With the same procedure as first case, one can 푔′′ ⟹ = −훼푘 obtain the following Theorem: (푔′)2 5 1 + 2 2 Theorem 4. The weighted minimal Monge hypersurface 1+(푘5) +(푘3) in Euclidean 4-space with linear density 푒훽푦 for (훽 ≠ 푔′′ 2 2 0) ∈ ℝ can be parametrized by √1+(푘5) +(푘3) −훼푘5 ⟹ ′ 2 = (푔 ) √1 + (푘 )2 + (푘 )2 푋(푥, 푦, 푧) = ( 푥, 푦, 푧, 푘5푥 + 푘1푦 + 푘 + 1 + 2 2 5 3 1+(푘5) +(푘3) 훽푘1푧 2 2 푙푛(푐표푠( +휆7))(1+(푘5) +(푘1) ) ′ 2 2 푔 √1+(푘5) +(푘1) ⟹ 푎푟푐푡푎푛 ( ) ), (3.17) 2 2 훽푘1 √1 + (푘5) + (푘3) 푋(푥, 푦, 푧)=( 푥, 푦, 푧, 푘1푦 + 푘3푧 + 푙 + −훼푘5푦 = + 휆3 훽푘1푥 2 2 √1+(푘 )2+(푘 )2 푙푛(푐표푠( +휆9))(1+(푘3) +(푘1) ) 5 3 2 2 √1+(푘3) +(푘1) −훼푘 푦 ) (3.18) ′ 2 2 5 훽푘 ⟹ 푔 = √1 + (푘5) + (푘3) 푡푎푛 ( + 휆3) 1 2 2 √1 + (푘5) + (푘3)
210 MONGE HYPERSURFACES IN EUCLIDEAN 4-SPACE WITH DENSITY- … Politeknik Dergisi, 2020; 23 (1) : 207-214
or 푋(푥, 푦, 푧) =( 푥, 푦, 푧, 푘5푥 + 푘3푧 + 푙 − 푦휇 2 2 푋(푥, 푦, 푧)=( 푥, 푦, 푧, 푘5푥 + 푘3푧 + 푛 + 푙푛(푐표푠( +휆21))(1+(푘5) +(푘1) ) 2 2 2 2 −휆 2훽푦 2휆 √1+(푘5) +(푘1) √(1+(푘5) +(푘3) )푎푟푐푡푎푛(푒 11√푒 −푒 11) ) (3.26) ), (3.19) 휇 훽 or where 푘 = 푘6 + 푘2 + 휆8, 푙 = 푘2 + 푘4 + 휆10 and 푛 = 푘6 + 푘4 + 휆12. 푋(푥, 푦, 푧) = ( 푥, 푦, 푧, 푘5푥 + 푘1푦 + 푛 − 푧휇 2 2 훾푧 푙푛(푐표푠( +휆23))(1+(푘3) +(푘5) ) Case 3. Let the density be 푒 : 2 2 √1+(푘3) +(푘5) In this case, let us consider the Monge hypersurface ), (3.27) 휇 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 푚(푧)) where 푘 = 푘2 + 푘4 + 휆20, 푙 = 푘6 + 푘4 + 휆22 and 푛 = 훾푧 in Euclidean 4-space with linear density 푒 . Then, this 푘6 + 푘2 + 휆24. surface is weighted minimal if and only if 3.2. Weighted Flat Monge Hypersurfaces in 푬ퟒ with 0 = ℎ′′(푥)(1 + 푔′(푦)2 + 푚′(푧)2) Linear Density +푔′′(푦)(1 + ℎ′(푥)2 + 푚′(푧)2) (3.20) From (1.11), the weighted Gaussian curvature of the +푚′′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2) Monge hypersurface in Euclidean 4-space with linear density 푒훼푥+훽푦+훾푧+휇푡 is obtained as +훾푚′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2 + 푚′(푧)2) ℎ′′(푥)푔′′(푦)푚′′(푧) satisfies. Hence, from (3.20) we have 퐾휑 = − 3 . (3.28) (1+ℎ′(푥)2+푔′(푦)2+푚′(푧)2)2 Theorem 5. The weighted minimal Monge hypersurface So from (3.28), we can state the following theorems: in Euclidean 4-space with linear density 푒훾푧 for (훾 ≠ 0) ∈ ℝ can be parametrized by Theorem 7. Let 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 푚(푧)) be a Monge hypersurface in Euclidean 4-space 푋(푥, 푦, 푧) =( 푥, 푦, 푧, 푘1푦 + 푘3푧 + 푘 + 훼푥+훽푦+훾푧+휇푡 훾푘3푥 2 2 with linear density 푒 , where 훼, 훽, 훾 and 휇 are 푙푛(푐표푠( +휆13))(1+(푘3) +(푘1) ) 2 2 √1+(푘3) +(푘1) not all zero constants. If one of the functions ℎ(푥), 푔(푦) ), (3.21) and 푚 ( 푧 ) is linear, then 푀 is weighted flat. 훾푘3
푋(푥, 푦, 푧) =( 푥, 푦, 푧, 푘5푥 + 푘3푧 + 푙 + Theorem 8. If 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 훾푘3푦 2 2 ( )) 푙푛 (푐표푠 ( +휆15))(1+(푘3) +(푘5) ) 푚 푧 is a Monge hypersurface in Euclidean 4-space 2 2 √1+(푘3) +(푘1) with linear density 푒훼푥+훽푦+훾푧+휇푡, where 훼, 훽, 훾 and 휇 are ) (3.22) 훾푘3 not all zero constants, then its weighted Gaussian or curvature cannot be constant except for zero.
푋(푥, 푦, 푧)=( 푥, 푦, 푧, 푘5푥 + 푘1푦 + 푛+ 2 2 −휆 2훾푧 2휆 ퟒ √(1+(푘5) +(푘1) )푎푟푐푡푎푛(푒 17√푒 −푒 17) 4. MONGE HYPERSURFACES IN 푬 WITH ), (3.23) 휶풙ퟐ+휷풚ퟐ+휸풛ퟐ+흁풕ퟐ 훾 DENSITY 풆 where k= 푘2 + 푘4+휆14, l= 푘6 + 푘4 + 휆16 and 푛 = 푘6 + In this section, we obtain the weighted minimal Monge 푘2 + 휆18 . hypersurfaces and give a characterization for the Case 4. Let the density be 푒휇푡: constancy of weighted Gaussian curvature of Monge 4 훼푥2+훽푦2+훾푧2+휇푡2 Here, let us consider the Monge hypersurface hypersurfaces in 퐸 with density 푒 . 4.1. Weighted Minimal Monge Hypersurfaces in 푬ퟒ 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 푚(푧)) ퟐ ퟐ ퟐ ퟐ with Density 풆휶풙 +휷풚 +휸풛 +흁풕 in Euclidean 4-space with linear density 푒휇푡. Then, this surface is weighted minimal if and only if From (1.10) and (2.4), the weighted mean curvature of the Monge hypersurface 0 = ℎ′′(푥)(1 + 푔′(푦)2 + 푚′(푧)2) 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푓(푥, 푦, 푧)) +푔′′(푦)(1 + ℎ′(푥)2 + 푚′(푧)2) (3.24) 4 훼푥2+훽푦2+훾푧2+휇푡2 +푚′′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2) in 퐸 with density 푒 is obtained as −푓 (1+푓2+푓2)−푓 (1+푓2+푓2) −휇(1 + ℎ′(푥)2 + 푔′(푦)2 + 푚′(푧)2) 푥푥 푦 푧 푦푦 푥 푧 +푓 (1+푓2+푓2) satisfies. Thus, 푧푧 푥 푦 2(푓푥푦푓푥푓푦+푓푥푧푓푥푓푧+푓푦푧푓푦푓푧)− Theorem 6. The weighted minimal Monge hypersurface 2 2 2 {2(훼푥푓푥+훽푦푓푦+훾푧푓푧−휇푓)(1+푓푥 +푓푦 +푓푧 )} in Euclidean 4-space with linear density 푒휇푡 for (휇 ≠ 퐻 = . (4.1) 휑 2 2 2 3/2 0) ∈ 푅 can be parametrized by 3(1+푓푥 +푓푦 +푓푧 ) Thus, we get 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푘1푦 + 푘3푧 + 푘 − 푥휇 2 2 푙푛(푐표푠( +휆19))(1+(푘3) +(푘1) ) Proposition 3. Let 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푓(푥, 푦, 푧)) be 2 2 √1+(푘3) +(푘1) a Monge hypersurface in Euclidean 4-space with density , (3.25) 2 2 2 2 휇 푒훼푥 +훽푦 +훾푧 +휇푡 , where 훼, 훽, 훾 and 휇 are not all zero
211 Mustafa ALTIN, Ahmet KAZAN, H. Bayram KARADAĞ / POLİTEKNİK DERGİSİ, Politeknik Dergisi,2020;23(1): 207-214 constants. Then, this surface is weighted minimal if and From (4.6), we have only if ′′ 2 2 ℎ (1 + (푘1) + (푘3) ) 2(푓 푓 푓 + 푓 푓 푓 + 푓 푓 푓 )= ′ ′ 2 2 2 = −2훼푥 푥푦 푥 푦 푥푧 푥 푧 푦푧 푦 푧 ℎ (1 + (ℎ ) + (푘1) + (푘3) ) 2 2 2 2 2 2 ′′ ′′ ′ 푓푥푥(1 + 푓푦 + 푓푧 ) + 푓푦푦(1 + 푓푥 + 푓푧 )+푓푧푧(1 + 푓푥 + 푓푦 ) ℎ ℎ ℎ 2 2 2 ⟹ ′ − ′ 2 2 2 = −2훼푥 +(훼푥푓푥 + 훽푦푓푦 + 훾푧푓푧 − 휇푓)(1 + 푓푥 + 푓푦 + 푓푧 ) (4.2) ℎ (ℎ ) + 1 + (푘1) + (푘3) satisfies. 1 ′ ⟹ (ln|ℎ′| − ln|ℎ′2 + 1 + (푘 )2 + (푘 )2|) = −2훼푥 Here, if we take 푓(푥, 푦, 푧) = ℎ(푥) + 푔(푦) + 푚(푧), 2 1 3 2 ′ where ℎ, 푔 and 푚 are 퐶 −differentiable functions, then ℎ 2 using (3.3) in (4.2), the weighted mean curvature of the ⟹ ln | | = −훼푥 + 휆25 √(ℎ′)2 + 1 + (푘 )2 + (푘 )2 Monge hypersurface 1 3 ′ ℎ 2 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 푚(푧)) ⟹ = 푒−훼푥 +휆25 √(ℎ′)2 + 1 + (푘 )2 + (푘 )2 is obtained as 1 3 ′ −훼푥2+휆 ′ 2 2 2 ′′( ) ′ 2 ′ 2 ⟹ ℎ = 푒 25√(ℎ ) + 1 + (푘 ) + (푘 ) −ℎ 푥 (1+푔 (푦) +푚 (푧) ) 1 3 −푔′′(푦)(1+ℎ′(푥)2+푚′(푧)2) 2 ⟹ ℎ′ = 푒−훼푥 +휆25√(ℎ′)2 + 1 + (푘 )2 + (푘 )2 −푚′′(푧)(1+ℎ′(푥)2+푔′(푦)2) 1 3 ′ 2 −2훼푥2+2휆 ′ 2 2 2 −2(훼푥ℎ′(푥)+훽푦푔′(푦)+훾푧푚′(푧))− ⟹ (ℎ ) = 푒 25((ℎ ) + 1 + (푘 ) + (푘 ) ) 1 3 ′ 2 ′ 2 ′ 2 2 {휇(ℎ(푥)+푔(푦)+푚(푧))(1+ℎ (푥) +푔 (푦) +푚 (푧) )} 2 2 −훼푥 +휆25 퐻 = . (4.3) √1 + (푘1) + (푘3) . 푒 휑 3(1+ℎ′(푥)2+푔′(푦)2+푚′(푧)2)3/2 ⟹ ℎ′ = 2 2 Proposition 4. Let 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + √1 − (푒−훼푥 +휆25)
푔(푦) + 푚(푧)) be a Monge hypersurface in Euclidean 4- 2 √1+(푘 )2+(푘 )2.푒−훼푥 +휆25 훼푥2+훽푦2+훾푧2+휇푡2 1 3 space with density 푒 , where 훼, 훽, 훾 and ⟹ ℎ = ∫ 2 푑푥 . (4.7) −훼푥2+휆 휇 are not all zero constants. Then, this surface is √1−(푒 25) weighted minimal if and only if Thus, ′′ ′ 2 ′ 2 2 0 = ℎ (푥)(1 + 푔 (푦) + 푚 (푧) ) + 2 2 −훼푥2+푒−훼푥 +휆25 √1+(푘1) +(푘3) .푒 ′′ ′ 2 ′ 2 푓(푥, 푦, 푧) = ∫ 푑푥 + 푔 (푦)(1 + ℎ (푥) + 푚 (푧) )+ 2 2 √1−(푒−훼푥 +푛) 푚′′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2) + (4.4) 푘1푦 + 푘3푧 + 푘2 + 푘4 ′ ′ ′ 2(훼푥ℎ (푥) + 훽푦푔 (푦) + 훾푧푚 (푧)) − is a solution of (4.6). 휇(ℎ(푥) + 푔(푦) + 푚(푧))(1 + ℎ′(푥)2 + 푔′(푦)2 + 푚′(푧)2) So, we can give the following Theorem: satisfies. Theorem 9. The weighted minimal Monge hypersurface 2 Now, we’ll obtain the weighted minimal Monge in Euclidean 4-space with density 푒훼푥 for (훼 ≠ 0) ∈ ℝ 2 2 2 2 hypersurfaces in 퐸4 with density 푒훼푥 +훽푦 +훾푧 +휇푡 for can be parametrized by 2 different choices of the not all zero constants 훼, 훽, 훾 and √1+(푘 )2+(푘 )2.푒−훼푥 +휆25 푋(푥, 푦, 푧) =(푥, 푦, 푧, ∫ 1 3 푑x 휇. 2 2 √ ( −훼푥 +휆25) 2 1− 푒 Case 1. Let the density be 푒훼푥 : +푘 푦 + 푘 푧 + 푘 + 푘 ). (4.8) In this case, let us consider the Monge hypersurface 1 3 2 4 Secondly, taking the functions ℎ(푥) and 푚(푧) are linear, 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 푚(푧)) i.e. ℎ(푥) = 푘5푥 + 푘6, 푚(푧) = 푘3푧 + 푘4, from (4.5), we 2 in Euclidean 4-space with density 푒훼푥 . Then, this have ′′ 2 2 surface is weighted minimal if and only if 푔 (1 + (푘5) + (푘3) )= ′′ ′ 2 ′ 2 ′ 2 2 2 0 = ℎ (푥)(1 + 푔 (푦) + 푚 (푧) ) + −2훼푥푘5(1 + (푔 ) + (푘5) + (푘3) ). (4.9) ′′ ′ 2 ′ 2 ′′ 푔 (푦)(1 + ℎ (푥) + 푚 (푧) ) + The equation (4.9) satisfies for 푘5 = 0 and 푔 (푦) = 0. 푚′′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2) + Similarly, taking the functions ℎ(푥) and 푔(푦) are linear, 2훼푥ℎ′(푥)(1 + ℎ′(푥)2 + 푔′(푦)2 + 푚′(푧)2) (4.5) i.e. ℎ(푥) = 푘5푥 + 푘6, 푔(푦) = 푘1푦 + 푘2, from (4.5), we have satisfies. Here, by obtaining some special solutions for ′′ 2 2 the equation (4.5), we’ll construct the weighted minimal 푚 (1 + (푘5) + (푘1) )= 4 훼푥2 ′ 2 2 2 Monge hypersurfaces in 퐸 with density 푒 . −2훼푥푘5(1 + (푚 ) + (푘5) + (푘1) ). (4.10) ′′ Firstly, let us take the functions 푔(푦) and 푚(푧) are linear, The equation (4.10) satisfies for 푘5 = 0 and 푚 (푧) = 0. i.e. 푔(푦) = 푘1푦 + 푘2, 푚(푧) = 푘3푧 + 푘4. Then, the So, we get equation (4.5) becomes Theorem 10. The weighted minimal Monge hypersurface ′′ 2 2 2 ℎ (1 + (푘1) + (푘3) ) = in Euclidean 4-space with linear density 푒훼푥 for (훼 ≠ ′ ′ 2 2 2 −2훼푥ℎ (1 + (ℎ ) + (푘1) + (푘3) ). (4.6) 0) ∈ ℝ can be parametrized by
212 MONGE HYPERSURFACES IN EUCLIDEAN 4-SPACE WITH DENSITY- … Politeknik Dergisi, 2020; 23 (1) : 207-214
′′ ′ 2 ′ 2 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푘1푦 + 푘3푧 + 푘), (4.11) = ℎ (푥)(1 + 푔 (푦) + 푚 (푧) ) ′′ ′ 2 ′ 2 where 푘 = 푘2 + 푘4 + 푘6. +푔 (푦)(1 + ℎ (푥) + 푚 (푧) ) (4.18) 2 Case 2. Let the density be 푒훽푦 : +푚′′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2) In this case, let us consider the Monge hypersurface satisfies. 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 푚(푧)) 4.2. The Constancy of Weighted Gaussian Curvature of Monge Hypersurfaces in 푬ퟒ with Density 훽푦2 in Euclidean 4-space with linear density 푒 . Then, this ퟐ ퟐ ퟐ ퟐ 풆휶풙 +휷풚 +휸풛 +흁풕 surface is weighted minimal if and only if From (1.11), the weighted Gaussian curvature of the 0 = ℎ′′(푥)(1 + 푔′(푦)2 + 푚′(푧)2) + Monge hypersurface in Euclidean 4-space with density ′′ ′ 2 ′ 2 2 2 2 2 +푔 (푦)(1 + ℎ (푥) + 푚 (푧) ) (4.12) 푒훼푥 +훽푦 +훾푧 +휇푡 is obtained as ′′ ( ′ 2 ′ 2) +푚 (푧) 1 + ℎ (푥) + 푔 (푦) ℎ′′(푥)푔′′(푦)푚′′(푧)− ′ ′ 2 ′ 2 ′ 2 ( 3/2) +2훽푦푔 (푦)(1 + ℎ (푥) + 푔 (푦) + 푚 (푧) ) 2(훼+훽+훾+휇)(1+ℎ′(푥)2+푔′(푦)2+푚′(푧)2) 퐾휑 = 3/ . (4.19) satisfies. With the same procedure as first case, one can (1+ℎ′(푥)2+푔′(푦)2+푚′(푧)2) 2 obtain the following Theorem: So from (4.19), we can state the following Theorem: Theorem 11. The weighted minimal Monge hypersurface 2 Theorem 13. Let 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + in Euclidean 4-space with linear density 푒훽푦 for (훽 ≠ 0) ∈ ℝ can be parametrized by 푔(푦) + 푚(푧)) be a Monge hypersurface in Euclidean 4- 훼푥2+훽푦2+훾푧2+휇푡2 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푘 푥 + 푘 푧 + 푘 + 푘 + space with density 푒 , where 훼, 훽, 훾 and 5 3 4 6 휇 are not all zero constants. If one of the functions ℎ(푥), 2 2 −훽푦2+휆 √1+(푘5) +(푘3) .푒 26 −푟 ∫ 2 푑푦) (4.13) 푔(푦) and 푚(푧) is linear and 훼 + 훽 + 훾 + 휇 = , then −훽푦2+휆 2 √1−(푒 26) the weighted Gaussian curvature of 푀 is constant 푟. or 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푘5푥 + 푘3푧 + 푘), (4.14) 5. CONCLUSION where 푘 = 푘2 + 푘4 + 푘6. Surface theory has an important place in 4-dimensional 2 Case 3. Let the density be 푒훾푧 : spaces as in 3-dimensional spaces. So, in the peresent study, we consider the Monge hypersurfaces in Euclidean Here, let us consider the Monge hypersurface 4-space with different densities and obtain the weighted 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 푚(푧)) minimal and weighted flat Monge hypersurfaces in this 2 in Euclidean 4-space with linear density 푒훾푧 . Then, this space. We think that, the results which are obtained in surface is weighted minimal if and only if this study are important for differential geometers who ′′ ′ 2 ′ 2 are dealing with weighted surfaces and in the near future, 0 = ℎ (푥)(1 + 푔 (푦) + 푚 (푧) ) the results which are stated in this study can be handled + 푔′′(푦)(1 + ℎ′(푥)2 + 푚′(푧)2) (4.15) in different four or higher dimensional spaces. + 푚′′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2) +2훾푚′(푧)(1 + ℎ′(푥)2 + 푔′(푦)2 + 푚′(푧)2) REFERENCES satisfies. Hence, we have [1] Moore, C., “Surfaces of rotation in a space of four Theorem 12. The weighted minimal Monge hypersurface dimensions”, Ann. Math., 21(2): 81-93, (1919). 2 in Euclidean 4-space with linear density 푒훾푧 for (훾 ≠ [2] Moore, C., “Rotation surfaces of constant curvature 0) ∈ ℝ can be parametrized by in space of four dimensions”, Bull. Amer. Math. Soc., 26(10): 454-460, (1920). 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푘5푥 + 푘1푦 + 푘2 + 푘6 2 [3] Cheng Q.M.and Wan, Q.R., “Complete √1+(푘 )2+(푘 )2.푒−훾푧 +휆27 +∫ 5 1 푑푥) (4.16) hypersurfaces of 푅4 with constant mean curvature”, 2 2 √1−(푒−훾푧 +휆27) Monatsh. Mth., 118(3-4): 171-204, (1994). or [4] Yoon, D.W., “Rotation surfaces with finite type 4 푋(푥, 푦, 푧) = (푥, 푦, 푧, 푘 푥 + 푘 푦 + 푘), (4.17) Gauss map in 퐸 ”, Indian J. Pure Appl. Math., 5 1 32(12): 1803-1808, (2001). where 푘 = 푘 + 푘 + 푘 . 2 4 6 [5] Arslan, K., Kılıç, B, Bulca, B. and Öztürk, G., 휇푡2 Case 4. Let the density be 푒 : “Generalized Rotation Surfaces in 퐸4”, Results In this case, let us consider the Monge hypersurface Math., 61(3-4): 315-327, (2012). 푀: 푋(푥, 푦, 푧) = (푥, 푦, 푧, ℎ(푥) + 푔(푦) + 푚(푧)) [6] Arslan, K., Bayram, B., Bulca, B. and Öztürk, G., 휇푡2 “On translation surfaces in 4-dimensional in Euclidean 4-space with linear density 푒 . Then, this Euclidean space”, Acta et Com. Uni. Tar. De surface is weighted minimal if and only if Math., 20(2):123-133, (2016). 2휇(ℎ(푥) + 푔(푦) + 푚(푧))(1 + ℎ′(푥)2 + 푔′(푦)2 + 푚′(푧)2)
213 Mustafa ALTIN, Ahmet KAZAN, H. Bayram KARADAĞ / POLİTEKNİK DERGİSİ, Politeknik Dergisi,2020;23(1): 207-214
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