Constructions of helicoidal surfaces by using curvature functions in isotropic space

Year 2019, Volume: 48 Issue: 4, 959 - 965, 08.08.2019

Dae Won Yoon , Jae Won Lee , Chul Woo Lee

Abstract

In the present  paper, we study helicoidal surfaces in the three dimensional isotropic space $\Bbb I^3$ and construct  helicoidal surfaces with prescribed Gaussian curvature or mean curvature given by smooth functions. Moreover, we give some examples of helicoidal surfaces with non-constant Gaussian curvature or mean curvature.

Keywords

Isotropic space, helicoidal surface, mean curvature, Gaussian curvature

References

• [1] M.E. Aydin, Classification results on surfaces in the isotropic 3-space, AKU J. Sci. Eng. 16, 239-246, 2016.
• [2] C. Baikoussis and T. Koufogiorgos, T., Helicoidal surfaces with prescribed mean or Gaussian curvature, J. Geom. 63, 25-29, 1988.
• [3] Chr.C. Beneki, G. Kaimakamis and B.J. Papantonios, Helicoidal surfaces in threedimensional Minkowski space, J. Math. Anal. Appl. 275, 586-614, 2002.
• [4] R. Caddeo, P. Piu and A. Ratto, Rotation surfaces in $H_3$ with constant Gauss curvature, Bollettino U.M.I. 7, 341-357, 1996.
• [5] C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante, J. Math. Pures Appl. Series 1 6, 309-320, 1841.
• [6] M.P. do Carmo and M. Dajczer, Helicoidal surfaces with constant mean curvature, Tôhoku Math. J. 34 (3), 425-435, 1982.
• [7] A. Gray, Modern differential geometry of curves and surfaces, CRC Press 1993.
• [8] J.-I. Hano and K. Nomizu, Surfaces of revolution with constant mean curvature in Lorentz-Minkowski space, Tôhoku Math. J. 36, 427-437, 1984.
• [9] F. Ji and Z.H. Hou, Helicoidal surfaces under the cubic screw motion in Minkowski 3-space, J. Math. Anal. Appl. 318, 634-647, 2006.
• [10] H. Pottmann, P. Grohs and N.J. Mitra, Laguerre minimal surfaces, isotropic geometry and linear elasticity, Adv. Comput. Math. 31, 391-419, 2009.
• [11] Ž.M. Šipuš, Translation surfaces of constant curvatures in a simply isotropic space, Period. Math. Hungar. 68, 160-175, 2014.
• [12] D.W. Yoon, D.-S. Kim, Y.H. Kim and J.W. Lee, Helicoidal surfaces with prescribed curvature in $Nil_3$, International J. Math. 24, 1350107 (11 pages), 2013.