On helicoidal surfaces in a conformally flat 3-space. (English) Zbl 1377.53025
Summary: In this paper, we discuss the problem of finding explicit parametrizations for the helicoidal surfaces in a conformally flat 3-space \(\mathbb {E}^3_F\) with prescribed extrinsic curvature or mean curvature given by smooth functions. Also, we give examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in \(\mathbb {E}^3_F\).
References:
| [1] | Araújo, K.O., Cui, N., Pina, R.S.: Helicoidal minimal surfaces in a conformally flat \[33\]-space. Bull. Korean Math. Soc. 53, 531-540 (2016) · Zbl 1342.53083 · doi:10.4134/BKMS.2016.53.2.531 |
| [2] | Baikoussis, C., Koufogiorgos, T.: Helicoidal surfaces with prescribed mean or Gaussian curvature. J. Geom. 63, 25-29 (1998) · Zbl 0943.53004 · doi:10.1007/BF01221235 |
| [3] | Beneki, C.C., Kaimakamis, G., Papantonios, B.J.: Helicoidal surfaces in three-dimensional Minkowski space. J. Math. Anal. Appl. 275, 586-614 (2002) · Zbl 1022.53016 · doi:10.1016/S0022-247X(02)00269-X |
| [4] | Corro, A.V., Pina, R., Souza, M.: Surfaces of revolution with constant extrinsic curvature in a conformally flat \[33\]-space. Results Math. 60, 225-234 (2011) · Zbl 1251.53037 · doi:10.1007/s00025-011-0172-3 |
| [5] | Ji, F., Hou, Z.H.: Helicoidal surfaces under the cubic screw motion in Minkowski \[33\]-space. J. Math. Anal. Appl. 318, 634-647 (2006) · Zbl 1094.53009 · doi:10.1016/j.jmaa.2005.06.032 |
| [6] | Kenmotsu, K.: Surfaces of revolution with prescribed mean curvature. Tôhoku Math. J. 32, 147-153 (1980) · Zbl 0431.53005 · doi:10.2748/tmj/1178229688 |
| [7] | Yoon, D.W., Kim, D.-S., Kim, Y.H., Lee, J.W.: Helicoidal surfaces with prescribed curvatures in \[Nil_3\] Nil3. Int. J. Math. 24, 1350107 (2013). (11 pages) · Zbl 1284.53058 · doi:10.1142/S0129167X13501073 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
