Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space

Georgi Ganchev; Velichka Milousheva

Open Mathematics (2014)

  • Volume: 12, Issue: 10, page 1586-1601
  • ISSN: 2391-5455

Abstract

top
In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all rotational quasi-minimal surfaces of elliptic, hyperbolic and parabolic type, respectively.

How to cite

top

Georgi Ganchev, and Velichka Milousheva. "Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space." Open Mathematics 12.10 (2014): 1586-1601. <http://eudml.org/doc/269222>.

@article{GeorgiGanchev2014,
abstract = {In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all rotational quasi-minimal surfaces of elliptic, hyperbolic and parabolic type, respectively.},
author = {Georgi Ganchev, Velichka Milousheva},
journal = {Open Mathematics},
keywords = {Pseudo-Euclidean 4-space with neutral metric; Quasi-minimal surfaces; Lightlike mean curvature vector; Rotational surfaces of elliptic; hyperbolic or parabolic type; pseudo-Euclidean 4-space with neutral metric; quasi-minimal surfaces; light-like mean curvature vector; rotational surfaces of elliptic},
language = {eng},
number = {10},
pages = {1586-1601},
title = {Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space},
url = {http://eudml.org/doc/269222},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Georgi Ganchev
AU - Velichka Milousheva
TI - Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space
JO - Open Mathematics
PY - 2014
VL - 12
IS - 10
SP - 1586
EP - 1601
AB - In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all rotational quasi-minimal surfaces of elliptic, hyperbolic and parabolic type, respectively.
LA - eng
KW - Pseudo-Euclidean 4-space with neutral metric; Quasi-minimal surfaces; Lightlike mean curvature vector; Rotational surfaces of elliptic; hyperbolic or parabolic type; pseudo-Euclidean 4-space with neutral metric; quasi-minimal surfaces; light-like mean curvature vector; rotational surfaces of elliptic
UR - http://eudml.org/doc/269222
ER -

References

top
  1. [1] Chen, B.-Y., Classification of marginally trapped Lorentzian flat surfaces in 𝔼 2 4 and its application to biharmonic surfaces, J. Math. Anal. Appl., 2008, 340(2), 861–875. http://dx.doi.org/10.1016/j.jmaa.2007.09.021 Zbl1160.53007
  2. [2] Chen, B.-Y., Classification of marginally trapped surfaces of constant curvature in Lorentzian complex plane, Hokkaido Math. J., 2009, 38(2), 361–408. http://dx.doi.org/10.14492/hokmj/1248190082 Zbl1187.53056
  3. [3] Chen, B.-Y., Black holes, marginally trapped surfaces and quasi-minimal surfaces. Tamkang J. Math., 2009, 40(4), 313–341. Zbl1194.53060
  4. [4] Chen, B.-Y., Pseudo-Riemannian geometry, δ-invariants and applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. http://dx.doi.org/10.1142/8003 
  5. [5] Chen, B.-Y., Dillen, F., Classification of marginally trapped Lagrangian surfaces in Lorentzian complex space forms, J. Math. Phys., 2007, 48(1), 013509, 23 pp.; Erratum, J. Math. Phys., 2008, 49(5), 059901, 1p. http://dx.doi.org/10.1063/1.2424553 Zbl1152.81370
  6. [6] Chen, B.-Y., Garay, O., Classification of quasi-minimal surfaces with parallel mean curvature vector in pseudo-Euclidean 4-space 𝔼 2 4 , Result. Math., 2009, 55(1–2), 23–38. http://dx.doi.org/10.1007/s00025-009-0386-9 Zbl1178.53049
  7. [7] Chen, B.-Y., Mihai, I., Classification of quasi-minimal slant surfaces in Lorentzian complex space forms, Acta Math. Hungar., 2009, 122(4), 307–328. http://dx.doi.org/10.1007/s10474-008-8033-6 Zbl1199.53167
  8. [8] Chen, B.-Y., Van der Veken, J., Marginally trapped surfaces in Lorentzian space with positive relative nullity, Class. Quantum Grav., 2007, 24(3), 551–563. http://dx.doi.org/10.1088/0264-9381/24/3/003 Zbl1141.53065
  9. [9] Chen, B.-Y., Van der Veken, J., Spatial and Lorentzian surfaces in Robertson-Walker space times, J. Math. Phys., 2007, 48(7), 073509, 12 pp. http://dx.doi.org/10.1063/1.2748616 Zbl1144.81324
  10. [10] Chen, B.-Y., Van der Veken, J., Classification of marginally trapped surfaces with parallel mean curvature vector in Lorenzian space forms, Houston J. Math., 2010, 36(2), 421–449. Zbl1213.53026
  11. [11] Chen, B.-Y., Yang, D., Addendum to “Classification of marginally trapped Lorentzian flat surfaces in 𝔼 2 4 and its application to biharmonic surfaces,” J. Math. Anal. Appl., 2010, 361(1), 280–282. http://dx.doi.org/10.1016/j.jmaa.2009.08.061 Zbl1179.53021
  12. [12] Ganchev, G., Milousheva, V., Chen rotational surfaces of hyperbolic or elliptic type in the four-dimensional Minkowski space, C. R. Acad. Bulgare Sci., 2011, 64(5), 641–652. Zbl1289.53014
  13. [13] Ganchev, G., Milousheva, V., An invariant theory of marginally trapped surfaces in the four-dimensional Minkowski space, J. Math. Phys., 2012, 53(3), 033705, 15 pp. http://dx.doi.org/10.1063/1.3693976 Zbl1274.83073
  14. [14] Haesen, S., Ortega, M., Boost invariant marginally trapped surfaces in Minkowski 4-space, Classical Quantum Gravity, 2007, 24(22), 5441–5452. http://dx.doi.org/10.1088/0264-9381/24/22/009 Zbl1165.83334
  15. [15] Haesen, S., Ortega, M., Marginally trapped surfaces in Minkowski 4-space invariant under a rotational subgroup of the Lorentz group, Gen. Relativity Gravitation, 2009, 41(8), 1819–1834. http://dx.doi.org/10.1007/s10714-008-0754-x Zbl1177.83017
  16. [16] Haesen, S., Ortega, M., Screw invariant marginally trapped surfaces in Minkowski 4-space, J. Math. Anal. Appl., 2009, 355(2), 639–648. http://dx.doi.org/10.1016/j.jmaa.2009.02.019 Zbl1166.83006
  17. [17] Liu H., Liu G., Hyperbolic rotation surfaces of constant mean curvature in 3-de Sitter space, Bull. Belg. Math. Soc. Simon Stevin, 2000, 7(3), 455–466. Zbl0973.53047
  18. [18] Liu H., Liu G., Weingarten rotation surfaces in 3-dimensional de Sitter space, J. Geom., 2004, 79(1–2), 156–168. http://dx.doi.org/10.1007/s00022-003-1567-4 Zbl1062.53060
  19. [19] Penrose, R. Gravitational collapse and space-time singularities, Phys. Rev. Lett., 1965, 14, 57–59. http://dx.doi.org/10.1103/PhysRevLett.14.57 Zbl0125.21206

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.