ERHAN GÜLER

Rotational hypersurfaces satisfying ∆ ?? = ?? in the four-dimensional Euclidean space

In this study, rotational hypersurfaces in the 4-dimensional Euclidean space are discussed. Some relations of curvatures of hypersurfaces are given, such as the mean, Gaussian, and their minimality and flatness. In addition, Laplace-Beltrami operator has been defined for 4-dimensional hypersurfaces depending on the first fundamental form. Moreover, it is shown that each element of the 4 × 4 order matrix ?, which satisfies the condition ∆ ?? = ??, is zero, that is, the rotational hypersurface ? is minimal.

Dört-Boyutlu Öklid Uzayında ∆ ?? = ?? Koşulunu Sağlayan Dönel Hiperyüzeyler

Bu çalışmada, 4-boyutlu Öklid uzayındaki dönel hiperyüzeyler ele alınmıştır. Hiperyüzeylerin ortalama, Gauss eğrilikleri hesaplanıp aralarındaki minimal ve düzlemsel olma durumları gibi bazı bağıntılar verilmiştir. Ayrıca, 4-boyutlu hiperyüzeyler için birinci temel forma bağlı olarak Laplace-Beltrami operatörü tanımlanmıştır. Üstelik, dönel yüzeyin ∆ ?? = ?? koşulunu sağlayan 4 × 4 mertebeli ? matrisinin her elemanının sıfır olduğu, yani ? dönel hiperyüzeyinin minimal olduğu gösterildi.

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[1] Altın M., Kazan A., Karadağ H.B., “Monge hypersurfaces in Euclidean 4-space with density,” J. Polytechnic, 23(1): 207–214, (2020).

[2] Arslan K., Kılıç Bayram B., Bulca B., Öztürk G., “Generalized Rotation Surfaces in ? 4 ,” Result Math. 61: 315–327, (2012).

[3] Arvanitoyeorgos A., Kaimakamis G., Magid M., “Lorentz hypersurfaces in ? 4,1 satisfying Δ? = ??,” Illinois J. Math. 53(2): 581–590, (2009).

[4] Bour E., “Théorie de la déformation des surfaces,” J. Êcole Imperiale Polytech., vol. 22, no. 39: 1–148, (1862).

[5] Chen B.Y., “Total mean curvature and submanifolds of finite type,” World Scientific, Singapore, (1984).

[6] Chen B.Y., Choi M., Kim Y.H., “Surfaces of revolution with pointwise 1-type Gauss map,” Korean Math. Soc., 42: 447–455, (2005).

[7] Cheng Q.M., Wan Q.R., “Complete hypersurfaces of ℝ4 with constant mean curvature,” Monatsh. Math. 118(3): 171–204, (1994).

[8] Choi M., Kim Y.H., “Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map,” Bull. Korean Math. Soc., 38: 753–761, (2001).

[9] Dillen F., Pas J., Verstraelen L., “On surfaces of finite type in Euclidean 3-space,” Kodai Math. J., 13: 10–21, (1990).

[10] Do Carmo M., Dajczer M., “Helicoidal surfaces with constant mean curvature,” Tohoku Math. J., 34: 351– 367, (1982).

[11] Dursun U., Turgay N.C., “Minimal and PseudoUmbilical Rotational Surfaces in Euclidean Space ? 4 ,” Mediter. J. Math. 10: 497–506, (2013).

[12] Ferrandez A., Garay O.J., Lucas P., “On a certain class of conformally at Euclidean hypersurfaces,” In Global Analysis and Global Differential Geometry; Springer: Berlin, Germany, 48–54. (1990).

[13] Ganchev G., Milousheva V., “General rotational surfaces in the 4-dimensional Minkowski space,” Turkish J. Math. 38: 883–895, (2014).

[14] Güler E., “Bour's theorem and lightlike profile curve,” Yokohama Math. J., 54(1): pp. 55–77, (2007).

[15] Güler E., Hacısalihoğlu H.H., Kim Y.H., The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-Space. Symmetry, 10(9): 398, 1–11 (2018).

[16] Güler E., Magid M., Yaylı Y., “Laplace Beltrami operator of a helicoidal hypersurface in four space,” J. Geom. Symm. Phys. 41: 77–95, (2016).

[17] Lawson H.B., “Lectures on Minimal Submanifolds,” 2nd ed.; Mathematics Lecture Series 9; Publish or Perish, Inc.: Wilmington, Delaware, (1980).

[18] Magid M., Scharlach C., Vrancken L., “Affine umbilical surfaces in ℝ4 ,” Manuscripta Math., 88: 275–289, (1995).

[19] Moore C., “Surfaces of rotation in a space of four dimensions,” Ann. Math., 21: 81–93, (1919).

[20] Moore C., “Rotation surfaces of constant curvature in space of four dimensions,” Bull. Amer. Math. Soc., 26: 454–460, (1920).

[21] Moruz M., Munteanu M.I., “Minimal translation hypersurfaces in ? 4 ,” J. Math. Anal. Appl., 439: 798– 812, (2016).

[22] Scharlach C., “Affine geometry of surfaces and hypersurfaces in ℝ4 ,” In Symposium on the Differential Geometry of Submanifolds; Dillen, F., Simon, U., Vrancken, L., Eds.; Un. Valenciennes: Valenciennes, France, 124: 251–256, (2007).

[23] Senoussi B., Bekkar M., “Helicoidal surfaces with Δ ? ? = ?? in 3-dimensional Euclidean space,” Stud. Univ. Babeş-Bolyai Math., 60(3): 437–448, (2015).

[24] Takahashi T., “Minimal immersions of Riemannian manifolds,” J. Math. Soc. Japan, 18: 380–385, (1966).

[25] Verstraelen L., Walrave J., Yaprak Ş., “The minimal translation surfaces in Euclidean space,“ Soochow J. Math., 20(1): 77–82, (1994).

[26] Vlachos TH., “Hypersurfaces in ? 4 with harmonic mean curvature vector field,” Math. Nachr., (172): 145–169, (1995).