Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

Rafael López ; Esma Demir

Open Mathematics, Tome 12 (2014), p. 1349-1361 / Harvested from The Polish Digital Mathematics Library

Résumé

We classify all helicoidal non-degenerate surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is 0 or 1 and that the surface is ruled. If the generating curve is a Lorentzian circle, we prove that the only possibility is that the axis is spacelike and the center of the circle lies on the axis.

Détail

Publié le : 2014-01-01

Zbl 1295.53008

EUDML-ID : urn:eudml:doc:269080

Bibliographie

[1] Beneki C.C., Kaimakamis G., Papantoniou B.J., Helicoidal surfaces in three-dimensional Minkowski space, J. Math. Anal. Appl., 2002, 275(2), 586–614 http://dx.doi.org/10.1016/S0022-247X(02)00269-X | Zbl 1022.53016

[2] Dillen F., Kühnel W., Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math., 1999, 98(3), 307–320 http://dx.doi.org/10.1007/s002290050142 | Zbl 0942.53004

[3] Hano J., Nomizu K., On isometric immersions of the hyperbolic plane into the Lorentz-Minkowski space and the Monge-Ampére equation of a certain type, Math. Ann., 1983, 262(2), 245–253 http://dx.doi.org/10.1007/BF01455315 | Zbl 0507.53042

[4] Hano J., Nomizu K., Surfaces of revolution with constant mean curvature in Lorentz-Minkowski space, Tôhoku Math. J., 1984, 36(3), 427–437 http://dx.doi.org/10.2748/tmj/1178228808 | Zbl 0535.53002

[5] Hou Z.H., Ji F., Helicoidal surfaces with H 2 = K in Minkowski 3-space, J. Math. Anal. Appl., 2007, 325(1), 101–113 http://dx.doi.org/10.1016/j.jmaa.2006.01.017 | Zbl 1112.53009

[6] Ji F., Hou Z.H., A kind of helicoidal surfaces in 3-dimensional Minkowski space, J. Math. Anal. Appl., 2005, 304(2), 632–643 http://dx.doi.org/10.1016/j.jmaa.2004.09.065 | Zbl 1162.53304

[7] Ji F., Hou Z.H., Helicoidal surfaces under the cubic screw motion in Minkowski 3-space, J. Math. Anal. Appl., 2006, 318(2), 634–647 http://dx.doi.org/10.1016/j.jmaa.2005.06.032 | Zbl 1094.53009

[8] Kobayashi O., Maximal surfaces in the 3-dimensional Minkowski space L 3, Tokyo J. Math., 1983, 6(2), 297–309 http://dx.doi.org/10.3836/tjm/1270213872 | Zbl 0535.53052

[9] López F.J., López R., Souam R., Maximal surfaces of Riemann type in Lorentz-Minkowski space L3, Michigan Math. J., 2000, 47(3), 469–497 http://dx.doi.org/10.1307/mmj/1030132590 | Zbl 1029.53014

[10] López R., Timelike surfaces with constant mean curvature in Lorentz three-space, Tôhoku Math. J., 2000, 52(4), 515–532 http://dx.doi.org/10.2748/tmj/1178207753 | Zbl 0981.53051

[11] López R., Surfaces of constant Gauss curvature in Lorentz-Minkowski three-space, Rocky Mountain J. Math., 2003, 33(3), 971–993 http://dx.doi.org/10.1216/rmjm/1181069938 | Zbl 1062.53003

[12] López R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom., 2014 (in press), preprint available at http://arxiv.org/abs/0810.3351 | Zbl 1312.53022

[13] Mira P., Pastor J.A., Helicoidal maximal surfaces in Lorentz-Minkowski space, Monatsh. Math., 2003, 140(4), 315–334 http://dx.doi.org/10.1007/s00605-003-0111-9 | Zbl 1050.53014

[14] O’Neill B., Semi-Riemannian Geometry, Pure Appl. Math., 103, Academic Press, New York, 1983

[15] Sasahara N., Spacelike helicoidal surfaces with constant mean curvature in Minkowski 3-space, Tokyo J. Math., 2000, 23(2), 477–502 http://dx.doi.org/10.3836/tjm/1255958684 | Zbl 0981.53052

[16] Strubecker K., Differentialgeometrie III, Sammlung Göschen, 1180, Walter de Gruyter, Berlin, 1959

[17] Weinstein T., An Introduction to Lorentz Surfaces, de Gruyter Exp. Math., 22, Walter de Gruyter, Berlin, 1996 http://dx.doi.org/10.1515/9783110821635