Research Article
BibTex RIS Cite

Year 2019, Volume: 12 Issue: 1, 20 - 25, 27.03.2019

Önder Gökmen Yıldız Mahmut Akyiğit

Abstract

References

  • [1] Baba-Hamed, C. and Bekkar, M., Helicoidal surfaces in the three-dimensional Lorentz-Minkowski space satisfying4II ri = iri. J. Geom. 100 (2011), no. 1-2, 1.
  • [2] Baikoussis, C. and Koufogiorgos, T., Helicoidal surfaces with prescribed mean or Gaussian curvature. J. Geom. 63 (1998), no. 1, 25-29.
  • [3] Beneki, C.C., Kaimakamis, G. and Papantoniou, B.J., Helicoidal surfaces in three-dimensional Minkowski space. Journal of Mathematical Analysis and Applications, 275 (2002), no. 2, 586-614.
  • [4] Corro, A. V., Pina, R. and Souza, M., Surfaces of rotation with constant extrinsic curvature in a conformally flat 3-space. Results in Mathematics, 60 (2011), no. 1-4, 225.
  • [5] Corwin, I., Hoffman, N., Hurder, S., Šešum, V. and Xu, Y., Differential geometry of manifolds with density. Rose-Hulman Undergrad. Math. J., 7(2006), 1-15.
  • [6] Delaunay, C. H., Sur la surface de révolution dont la courbure moyenne est constante. Journal de mathématiques pures et appliquées (1841), 309-314.
  • [7] Do Carmo, M.P. and Dajczer, M., Helicoidal surfaces with constant mean curvature. Tohoku Mathematical Journal, Second Series 34 (1982), no. 3, 425-435.
  • [8] Hıeu, D.T. and Hoang, N.M., Ruled minimal surfaces in R3 with density ez. Pacific Journal of Mathematics, 243 (2009), no. 2, 277-285.
  • [9] Hou, Z.H. and Ji, F., Helicoidal surfaces with H2 = K in Minkowski 3-space. Journal of mathematical analysis and applications, 325 (2007), no. 1, 101-113.
  • [10] Ji, F. and Hou, Z.H., A kind of helicoidal surfaces in 3-dimensional Minkowski space. Journal of mathematical analysis and applications, 304 (2005), no.2 632-643.
  • [11] Ji, F. and Hou, Z.H., Helicoidal surfaces under the cubic screw motion in Minkowski 3-space. Journal of mathematical analysis and applications, 318 (2006), no. 2, 634-647.
  • [12] Morgan, F., Geometric measure theory: a beginner’s guide. Academic press, 2016.
  • [13] Morgan, F., Manifolds with density. Notices of the AMS, (2005), 853-858.
  • [14] Morgan, F., Myers’ theorem with density. Kodai Mathematical Journal, 29 (2006), no. 3, 455-461.
  • [15] Morgan, F., Manifolds with Density and Perelman’s Proof of the Poincaré Conjecture. American Mathematical Monthly, 116 (2009), no. 2, 134-142.
  • [16] Rafael, L. and Demir, E., Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature. Open Mathematics, 12 (2014), no: 9, 1349-1361.
  • [17] Rayón, P. and Gromov, M., Isoperimetry of waists and concentration of maps. Geometric and functional analysis, 13 (2003), no. 1, 178-215.
  • [18] Rosales, C., Cañete, A., Bayle, V. and Morgan, M., On the isoperimetric problem in Euclidean space with density. Calculus of Variations and Partial Differential Equations, 31 (2008), no. 1, 27-46.
  • [19] Roussos, I.M., The helicoidal surfaces as Bonnet surfaces. Tohoku Mathematical Journal, Second Series 40 (1988), no. 3, 485-490.
  • [20] Yıldız, Ö.G., Hızal, S. and Akyi˘ git, M., Type I+ Helicoidal Surfaces with PrescribedWeighted Mean or Gaussian Curvature in Minkowski Space with Density. Analele Universitatii" Ovidius" Constanta-Seria Matematica, 26 (2018), no. 3, 99-108.
  • [21] Yoon, D.W.,Weighted minimal translation surfaces in Minkowski 3-space with density. International Journal of Geometric Methods in Modern Physics, 14 (2017), no. 12, 1750178.
  • [22] Yoon, D.W., Kim, D.S., Kim, Y.H. and Lee, J.W., Constructions of Helicoidal Surfaces in Euclidean Space with Density. Symmetry 9 (2017), no. 9, 173.

Constructions of Type III+ Helicoidal Surfaces in Minkowski Space with Desity

Year 2019, Volume: 12 Issue: 1, 20 - 25, 27.03.2019

Önder Gökmen Yıldız Mahmut Akyiğit

Abstract

In this paper, we construct a helicoidal surface of type III+ with prescribed weighted mean
curvature and weighted Gaussian curvature in the Minkowski 3-space R^3_1 with a positive density
function. We get a result for minimal case. Also we give examples of helicoidal surface with
prescribed weighted mean curvature and Gaussian curvature.

Keywords

Minkowski space manifold with density weighted curvature helicoidal

References

  • [1] Baba-Hamed, C. and Bekkar, M., Helicoidal surfaces in the three-dimensional Lorentz-Minkowski space satisfying4II ri = iri. J. Geom. 100 (2011), no. 1-2, 1.
  • [2] Baikoussis, C. and Koufogiorgos, T., Helicoidal surfaces with prescribed mean or Gaussian curvature. J. Geom. 63 (1998), no. 1, 25-29.
  • [3] Beneki, C.C., Kaimakamis, G. and Papantoniou, B.J., Helicoidal surfaces in three-dimensional Minkowski space. Journal of Mathematical Analysis and Applications, 275 (2002), no. 2, 586-614.
  • [4] Corro, A. V., Pina, R. and Souza, M., Surfaces of rotation with constant extrinsic curvature in a conformally flat 3-space. Results in Mathematics, 60 (2011), no. 1-4, 225.
  • [5] Corwin, I., Hoffman, N., Hurder, S., Šešum, V. and Xu, Y., Differential geometry of manifolds with density. Rose-Hulman Undergrad. Math. J., 7(2006), 1-15.
  • [6] Delaunay, C. H., Sur la surface de révolution dont la courbure moyenne est constante. Journal de mathématiques pures et appliquées (1841), 309-314.
  • [7] Do Carmo, M.P. and Dajczer, M., Helicoidal surfaces with constant mean curvature. Tohoku Mathematical Journal, Second Series 34 (1982), no. 3, 425-435.
  • [8] Hıeu, D.T. and Hoang, N.M., Ruled minimal surfaces in R3 with density ez. Pacific Journal of Mathematics, 243 (2009), no. 2, 277-285.
  • [9] Hou, Z.H. and Ji, F., Helicoidal surfaces with H2 = K in Minkowski 3-space. Journal of mathematical analysis and applications, 325 (2007), no. 1, 101-113.
  • [10] Ji, F. and Hou, Z.H., A kind of helicoidal surfaces in 3-dimensional Minkowski space. Journal of mathematical analysis and applications, 304 (2005), no.2 632-643.
  • [11] Ji, F. and Hou, Z.H., Helicoidal surfaces under the cubic screw motion in Minkowski 3-space. Journal of mathematical analysis and applications, 318 (2006), no. 2, 634-647.
  • [12] Morgan, F., Geometric measure theory: a beginner’s guide. Academic press, 2016.
  • [13] Morgan, F., Manifolds with density. Notices of the AMS, (2005), 853-858.
  • [14] Morgan, F., Myers’ theorem with density. Kodai Mathematical Journal, 29 (2006), no. 3, 455-461.
  • [15] Morgan, F., Manifolds with Density and Perelman’s Proof of the Poincaré Conjecture. American Mathematical Monthly, 116 (2009), no. 2, 134-142.
  • [16] Rafael, L. and Demir, E., Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature. Open Mathematics, 12 (2014), no: 9, 1349-1361.
  • [17] Rayón, P. and Gromov, M., Isoperimetry of waists and concentration of maps. Geometric and functional analysis, 13 (2003), no. 1, 178-215.
  • [18] Rosales, C., Cañete, A., Bayle, V. and Morgan, M., On the isoperimetric problem in Euclidean space with density. Calculus of Variations and Partial Differential Equations, 31 (2008), no. 1, 27-46.
  • [19] Roussos, I.M., The helicoidal surfaces as Bonnet surfaces. Tohoku Mathematical Journal, Second Series 40 (1988), no. 3, 485-490.
  • [20] Yıldız, Ö.G., Hızal, S. and Akyi˘ git, M., Type I+ Helicoidal Surfaces with PrescribedWeighted Mean or Gaussian Curvature in Minkowski Space with Density. Analele Universitatii" Ovidius" Constanta-Seria Matematica, 26 (2018), no. 3, 99-108.
  • [21] Yoon, D.W.,Weighted minimal translation surfaces in Minkowski 3-space with density. International Journal of Geometric Methods in Modern Physics, 14 (2017), no. 12, 1750178.
  • [22] Yoon, D.W., Kim, D.S., Kim, Y.H. and Lee, J.W., Constructions of Helicoidal Surfaces in Euclidean Space with Density. Symmetry 9 (2017), no. 9, 173.
There are 22 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Önder Gökmen Yıldız

Mahmut Akyiğit

Publication Date March 27, 2019
Published in Issue Year 2019 Volume: 12 Issue: 1

Cite

APA Yıldız, Ö. G., & Akyiğit, M. (2019). Constructions of Type III+ Helicoidal Surfaces in Minkowski Space with Desity. International Electronic Journal of Geometry, 12(1), 20-25.
AMA Yıldız ÖG, Akyiğit M. Constructions of Type III+ Helicoidal Surfaces in Minkowski Space with Desity. Int. Electron. J. Geom. March 2019;12(1):20-25.
Chicago Yıldız, Önder Gökmen, and Mahmut Akyiğit. “Constructions of Type III+ Helicoidal Surfaces in Minkowski Space With Desity”. International Electronic Journal of Geometry 12, no. 1 (March 2019): 20-25.
EndNote Yıldız ÖG, Akyiğit M (March 1, 2019) Constructions of Type III+ Helicoidal Surfaces in Minkowski Space with Desity. International Electronic Journal of Geometry 12 1 20–25.
IEEE Ö. G. Yıldız and M. Akyiğit, “Constructions of Type III+ Helicoidal Surfaces in Minkowski Space with Desity”, Int. Electron. J. Geom., vol. 12, no. 1, pp. 20–25, 2019.
ISNAD Yıldız, Önder Gökmen - Akyiğit, Mahmut. “Constructions of Type III+ Helicoidal Surfaces in Minkowski Space With Desity”. International Electronic Journal of Geometry 12/1 (March 2019), 20-25.
JAMA Yıldız ÖG, Akyiğit M. Constructions of Type III+ Helicoidal Surfaces in Minkowski Space with Desity. Int. Electron. J. Geom. 2019;12:20–25.
MLA Yıldız, Önder Gökmen and Mahmut Akyiğit. “Constructions of Type III+ Helicoidal Surfaces in Minkowski Space With Desity”. International Electronic Journal of Geometry, vol. 12, no. 1, 2019, pp. 20-25.
Vancouver Yıldız ÖG, Akyiğit M. Constructions of Type III+ Helicoidal Surfaces in Minkowski Space with Desity. Int. Electron. J. Geom. 2019;12(1):20-5.
 Download Cover Image