Mustafa ALTIN, Ahmet KAZAN, H. Bayram KARADAĞ

Monge Hypersurfaces in Euclidean 4-Space with Density

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.

Anahtar Kelimeler:

Manifold with density, weighted mean curvature, weighted gaussian curvature, monge hypersurfaces

Monge Hypersurfaces in Euclidean 4-Space with Density

Keywords:

Manifold with density, weighted mean curvature, weighted gaussian curvature, monge hypersurfaces,

PDF

___

[1] Moore, C., “Surfaces of rotation in a space of four dimensions”, Ann. Math., 21(2): 81-93, (1919).
[2] Moore, C., “Rotation surfaces of constant curvature in space of four dimensions”, Bull. Amer. Math. Soc., 26(10): 454-460, (1920).
[3] Cheng Q.M.and Wan, Q.R., “Complete hypersurfaces of R^4 with constant mean curvature”, Monatsh. Mth., 118(3-4): 171-204, (1994).
[4] Yoon, D.W., “Rotation surfaces with finite type Gauss map in E^4”, Indian J. Pure Appl. Math., 32(12): 1803-1808, (2001).
[5] Arslan, K., Kılıç, B, Bulca, B. and Öztürk, G., “Generalized Rotation Surfaces in E^4”, Results Math., 61(3-4): 315-327, (2012).
[6] Arslan, K., Bayram, B., Bulca, B. and Öztürk, G., “On translation surfaces in 4-dimensional Euclidean space”, Acta et Com. Uni. Tar. De Math., 20(2):123-133, (2016).
[7] Ganchev, G. and Milousheva, V., “General rotational surfaces in the 4-dimensional Minkowski space”, Turkish J. Math., 38: 883-895, (2014).
[8] Moruz, M. and Monteanu, M.I., “Minimal translation hypersurfaces in E^4”, J. Math. Anal. Appl. 439(2): 798-812, (2016).
[9] Dursun, U. and Turgay, N.C., “Minimal and pseudoumbilical rotational surfaces in Euclidean space E^4”, Mediterr. J. Math., 10(1): 497-506, (2013).
[10] Kahraman, F. and Yaylı, Y., “Boost invariant surface with pointwise 1-type Gauss map in Minkowski 4-space E^4_1”, Bull. Korean Math. Soc., 51: 1863-1874,(2014).
[11] Kahraman, F. and Yaylı, Y., “General rotational surfaces with pointwise 1-type Gauss map in pseudo-Euclidean space E^4_2”, Indian J. Pure Appl. Math., 46:107-118, (2014).
[12] Güler, E., Magid, M. and Yaylı, Y., “Laplace-Beltrami operator of a helicoidal hypersurface in four space”, J. Goem. and Sym. Phys., 41: 77-95, (2016).
[13] Güler, E., Hacısalihoμglu, H.H. and Kim, Y.H., “The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-space”, Symmetry, 10(398): 1-11, (2018).
[14] M. Gromov, “Isoperimetry of waists and concentration of maps”, Geom. Func. Anal., 13: 178-215, (2003).
[15] I. Corwin, N. Hoffman, S. Hurder, V. Sesum and Y. Xu, “Differential geometry of manifolds with density”, Rose-Hulman Und. Math. J., 7(1): 1-15, (2006).
[16] L. Belarbi and M. Belkhelfa, “Surfaces in R^3 with Density”, i-manager.s Journal on Mathematics, 1(1): 34-48, (2012).
[17] D.T. Hieu and T.L. Nam, “The classification of constant weighted curvature curves in the plane with a log-linear density”, Commun. Pure Appl. Anal., 13(4): 1641-1652, (2014).
[18] M. Altın, A. Kazan and H.B. Karadağ, “Rotational surfaces Generated by Planar Curves in E^3 with Density”, International Journal of Analysis and Applications, 17(3): 311-328, (2019).
[19] A. Kazan and H.B. Karadağ, “Weighted Minimal And Weighted Flat Surfaces of Revolution in Galilean 3-Space with Density”, Int. J. Anal. Appl., 16(3): 414-426, (2018).
[20] F. Morgan, “Manifolds with Density”, Not. Amer. Math. Soc., 52(8): 853-858, (2005).
[21] F. Morgan, “Myers’ Theorem With Density”, Kodai Math. J., 29: 455-461, (2006).
[22] T.L. Nam, “Some results on curves in the plane with log-linear density”, Asian-European J. of Math., 10(2): 1-8, (2017).
[23] D.W. Yoon, D-S. Kim, Y.H. Kim and J.W. Lee, “Constructions of Helicoidal Surfaces in Euclidean Space with Density”, Symmetry, 173: 1-9, (2017).
[24] D.W. Yoon and Z.K. Yüzbaşı, “Weighted Minimal Affine Translation Surfaces in Euclidean Space with Density”, International Journal of Geometric Methodsin Modern Physics, 15(11), 2018.
[25] Ö.G. Yıldız, S. Hızal and M. Akyiğit, “Type I+ Helicoidal Surfaces with Prescribed Weighted Mean or Gaussian Curvature in Minkowski Space with Density”,An. S.t. Univ. Ovidius Constanta, 26(3): 99-108, (2018).