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Timelike surfaces with zero mean curvature in Minkowski 4-space. (English) Zbl 1281.53022

The local theory of time-like surfaces with zero mean curvature in the 4-dimensional Minkowski space \(\mathbb R_1^4\) is studied. Special isothermal (canonical) parameters are introduced on any time-like surface with zero mean curvature free of flat points. Using the canonical parameters and canonical directions of such a surface one introduces a moving frame field at each point of the surface. It is shown that the Frenet-type derivative formulas with respect to this geometric frame field determine two invariant functions \(\mu\) and \(\nu\). The fundamental result of the paper is that any time-like surface with zero mean curvature free of flat points is determined up to a motion in \(\mathbb R_1^4\) by two natural partial differential equations satisfied by the invariants \(\mu\) and \(\nu\) and conversely. Equivalently, these equations can be expressed in functions of the Gauss curvature \(K\) and the normal curvature \(\chi\) as unknown quantities.
The results are applied to the class of time-like rotational surfaces of Moore type. The meridian curves of all such surfaces are obtained. It is shown that for this class of surfaces the curvatures \(K\) and \(\chi\) are functions of only one variable. The respective surfaces determine a one-parameter family of solutions to the system of natural partial differential equations describing time-like surfaces with zero mean curvature. Finally, a survey of the background systems of partial differential equations is given for the cases of the surfaces with zero mean curvature in Euclidean space \(\mathbb R^4\) or Minkowski space \(\mathbb R_1^4\) in terms of the Gauss curvature \(K\) and the normal curvature \(\chi\).

MSC:

53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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[1] L. Alías and B. Palmer, Curvature properties of zero mean curvature surfaces in four-dimensional Lorenzian space forms, Mathematical Proceedings of the Cambridge Philosophical Society 124 (1998), 315–327. · Zbl 0947.53031
[2] R. de Azevedo Tribuzy and I. Guadalupe, Minimal immersions of surfaces into 4-dimensional space forms, Rendiconti del Seminario Matematico della Universitá di Padova 73 (1985), 1–13. · Zbl 0573.53036
[3] G. Ganchev and V. Milousheva, On the theory of surfaces in the four-dimensional Euclidean space, Kodai Mathematical Journal 31 (2008), 183–198. · Zbl 1165.53003
[4] G. Ganchev and V. Milousheva, Invariants and Bonnet-type theorem for surfaces in \(\mathbb{R}\)4, Central European Journal of Mathematics 8 (2010), 993–1008. · Zbl 1213.53010
[5] G. Ganchev and V. Milousheva, An invariant theory of spacelike surfaces in the fourdimensional Minkowski space, Mediterranean Journal of Mathematics 9 (2012), 267–294. · Zbl 1277.53019
[6] S. Haesen and M. Ortega, Boost invariant marginally trapped surfaces in Minkowski 4-space, Classical and Quantum Gravity 24 (2007), 5441–5452. · Zbl 1165.83334
[7] S. Haesen and M. Ortega, Marginally trapped surfaces in Minkowski 4-space invariant under a rotation subgroup of the Lorentz group, General Relativity and Gravitation 41 (2009), 1819–1834. · Zbl 1177.83017
[8] T. Itoh, Minimal surfaces in 4-dimensional Riemannian manifolds of constant curvature, Kodai Mathematical Journal 23 (1971), 451–458. · Zbl 0249.53045
[9] E. Lane, Projective Differential Geometry of Curves and Surfaces, University of Chicago Press, Chicago, 1932. · Zbl 0005.02501
[10] J. Little, On singularities of submanifolds of higher dimensional Euclidean spaces, Annali di Matematica Pura ed Applicata, IV Ser. 83 (1969), 261–335. · Zbl 0187.18903
[11] H. Liu and G. Liu, Hyperbolic rotation surfaces of constant mean curvature in 3-de Sitter space, Belgian Mathematical Society 7 (2000), 455–466. · Zbl 0973.53047
[12] H. Liu and G. Liu, Weingarten rotation surfaces in 3-dimensional de Sitter space, Journal of Geometry 79 (2004), 156–168. · Zbl 1062.53060
[13] V. Milousheva, General rotational surfaces in \(\mathbb{R}\)4 with meridians lying in two-dimensional planes, Comptes Rendus de l’Academie Bulgare des Sciences 63 (2010), 339–348. · Zbl 1224.53015
[14] C. Moore Surfaces of rotation in a space of four dimensions, The Annals of Mathematics, 2nd Ser. 21 (1919), 81–93. · JFM 47.0974.01
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