General rotational ξ−surfaces in Euclidean spaces

Arslan, Kadri; Bulca, Betul; YILMAZ, AYDIN

doi:10.3906/mat-2006-93

General rotational ξ−surfaces in Euclidean spaces

Kadri ARSLAN, (Bursa Uludağ Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, Bursa, Türkiye)

Yılmaz AYDIN, (Bursa Uludağ Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, Bursa, Türkiye)

Betül BULCA (Bursa Uludağ Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, Bursa, Türkiye)

Turkish Journal of Mathematics

0 0

Yıl: 2021 Cilt: 45 Sayı: 3 Sayfa Aralığı: 1287 - 1299 Metin Dili: İngilizce DOI: 10.3906/mat-2006-93 İndeks Tarihi: 30-06-2022

General rotational ξ−surfaces in Euclidean spaces

Öz:

The general rotational surfaces in the Euclidean 4-space R 4 was first studied by Moore (1919). The Vranceanu surfaces are the special examples of these kind of surfaces. Self-shrinker flows arise as special solution of the mean curvature flow that preserves the shape of the evolving submanifold. In addition, ξ−surfaces are the generalization of self-shrinker surfaces. In the present article we consider ξ−surfaces in Euclidean spaces. We obtained some results related with rotational surfaces in Euclidean 4−space R 4 to become self-shrinkers. Furthermore, we classify the general rotational ξ−surfaces with constant mean curvature. As an application, we give some examples of self-shrinkers and rotational ξ−surfaces in R 4 .

Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

[1] Abresch U, Langer J. The normalized curve shortening flow and homothetic solutions. Journal of Differential Geometry 1986; 23: 175-196.
[2] Aminov Yu. The Geometry of Submanifolds. London, UK: Gordon and Breach Science Publication, 2001.
[3] Anciaux H. Construction of Lagrangian self-similar solutions to the mean curvature flow in C n . Geometriae Dedicata 2006; 120: 37-48.
[4] Arezzo C, Sun J. Self-shrinkers for the mean curvature flow in arbitrary codimension. Mathematische Zeitschrift 2013; 274: 993-1027.
[5] Arslan K, Bayram B, Bulca B, Kim YH, Murathan C et al. Rotational embeddings in E 4 with pointwise 1-type Gauss map. Turkish Journal of Mathematics 2011; 35: 493-499.
[6] Arslan K, Bayram B, Bulca B, Öztürk G. General rotation surfaces in E 4 . Results in Mathematics 2012; 61 (3): 315-327.
[7] Arslan K, Bulca B, Kosova D. On generalized rotational surfaces in Euclidean spaces. Journal of the Korean Mathematical Society 2017; 54 (3): 999-1013.
[8] Castro I, Lerma AM. The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow. International Mathematics Research Notices 2014; 16: 1515-1527.
[9] Chen BY. Geometry of Submanifolds. New York, NY, USA: Dekker, 1973.
[10] Chen BY. Differential geometry of rectifying submanifolds. International Electronic Journal of Geometry 2016; 9 (2):1-8
[11] Chen BY. More on convolution of Riemannian manifolds. Beitrage zur Algebra und Geometrie 2003; 44: 9-24.
[12] Cheng QM, Hori H, Wei G. Complete Lagrangian self-shrinkers in R 4 . arXiv 2018; arXiv:1802.02396.
[13] Cheng QM, Peng Y. Complete self-shrinkers of the mean curvature flow. Calculus of Variations Partial Differential Equations 2015; 52: 497-506.
[14] Cheng QM, Wei G. Complete λ−hypersurfaces of the weighted volume-preserving mean curvature flow. arXiv 2015; arXiv:1403.3177.
[15] Coung DV. Surfaces of revolution with constant Gaussian curvature in four space. Asian-European Journal of Mathematics 2013; 6: 1350021.
[16] Dursun U, Turgay NC. General rotational surfaces in Euclidean space E 4 with pointwise 1-type Gauss map. Mathematical Communications 2012; 17: 71-81.
[17] Ganchev G, Milousheva V. On the theory of surfaces in the four dimensional Euclidean space. Kodai Mathematical Journal 2008; 31: 183-198.
[18] Joyse D, Lee Y, Tsui MP. Self-similar solutions and translating solutions for Lagrangian mean curvature flow. Journal of Differential Geometry 2010; 84: 127-161.
[19] Li H, Wang X. New characterizations of the Clifford torus as a Lagrangian self-shrinker. The Journal of Geometric Analysis 2017; 27: 1393-1412.
[20] Li X, Chang X. A rigidity theorem of ξ− submanifolds in C 2 . Geometriae Dedicata 2016; 185: 155-169.
[21] Li X, Li Z. Variational characterization of ξ−submanifolds in the Euclidean space R m+p . Annali di Mathematica Pura ed Applicata 2020; 199: 1491-1518.
[22] Moore C. Surfaces of rotations in a space of four dimensions. Annals of Mathematics 1919; 21: 81-93.
[23] Smoczyk K. Self-shrinkers of the mean curvature flow in arbitrary codimension. International Mathematics Research Notices 2005; 48: 1983-3004.
[24] Vranceanu G. Surfaces de rotation dans E 4 . Revue Roumaine de Mathematıques Pures et Appliquees 1977; 22: 857-862.
[25] Wong YC. Contributions to the theory of surfaces in 4-space of constant curvature. Transactions of the American Mathematical Society 1946; 59: 467-507.
[26] Yoon DW. Some properties of the Clifford torus as rotation surface. Indian Journal of Pure and Applied Mathematics 2003; 34: 907-915.

APA	Arslan K, YILMAZ A, Bulca B (2021). General rotational ξ−surfaces in Euclidean spaces . , 1287 - 1299. 10.3906/mat-2006-93
Chicago	Arslan Kadri,YILMAZ AYDIN,Bulca Betul General rotational ξ−surfaces in Euclidean spaces . (2021): 1287 - 1299. 10.3906/mat-2006-93
MLA	Arslan Kadri,YILMAZ AYDIN,Bulca Betul General rotational ξ−surfaces in Euclidean spaces . , 2021, ss.1287 - 1299. 10.3906/mat-2006-93
AMA	Arslan K,YILMAZ A,Bulca B General rotational ξ−surfaces in Euclidean spaces . . 2021; 1287 - 1299. 10.3906/mat-2006-93
Vancouver	Arslan K,YILMAZ A,Bulca B General rotational ξ−surfaces in Euclidean spaces . . 2021; 1287 - 1299. 10.3906/mat-2006-93
IEEE	Arslan K,YILMAZ A,Bulca B "General rotational ξ−surfaces in Euclidean spaces ." , ss.1287 - 1299, 2021. 10.3906/mat-2006-93
ISNAD	Arslan, Kadri vd. "General rotational ξ−surfaces in Euclidean spaces ". (2021), 1287-1299. https://doi.org/10.3906/mat-2006-93

APA	Arslan K, YILMAZ A, Bulca B (2021). General rotational ξ−surfaces in Euclidean spaces . Turkish Journal of Mathematics, 45(3), 1287 - 1299. 10.3906/mat-2006-93
Chicago	Arslan Kadri,YILMAZ AYDIN,Bulca Betul General rotational ξ−surfaces in Euclidean spaces . Turkish Journal of Mathematics 45, no.3 (2021): 1287 - 1299. 10.3906/mat-2006-93
MLA	Arslan Kadri,YILMAZ AYDIN,Bulca Betul General rotational ξ−surfaces in Euclidean spaces . Turkish Journal of Mathematics, vol.45, no.3, 2021, ss.1287 - 1299. 10.3906/mat-2006-93
AMA	Arslan K,YILMAZ A,Bulca B General rotational ξ−surfaces in Euclidean spaces . Turkish Journal of Mathematics. 2021; 45(3): 1287 - 1299. 10.3906/mat-2006-93
Vancouver	Arslan K,YILMAZ A,Bulca B General rotational ξ−surfaces in Euclidean spaces . Turkish Journal of Mathematics. 2021; 45(3): 1287 - 1299. 10.3906/mat-2006-93
IEEE	Arslan K,YILMAZ A,Bulca B "General rotational ξ−surfaces in Euclidean spaces ." Turkish Journal of Mathematics, 45, ss.1287 - 1299, 2021. 10.3906/mat-2006-93
ISNAD	Arslan, Kadri vd. "General rotational ξ−surfaces in Euclidean spaces ". Turkish Journal of Mathematics 45/3 (2021), 1287-1299. https://doi.org/10.3906/mat-2006-93