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Vol. 49 No. 4 (2023)
Serdica Mathematical Journal

Articles

Minimal timelike surfaces in the Lorentz–Minkowski 3-space and their canonical parameters

https://doi.org/10.55630/serdica.2023.49.301-316

Published 21-12-2023

  • Ognian Kassabov
    • ,
  • Velichka Milousheva

Ognian Kassabov
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Velichka Milousheva
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Abstract

We study minimal timelike surfaces in \(\mathbb R^3_1\) using a special Weierstrass-type formula in terms of holomorphic functions defined in the algebra of the double (split-complex) numbers. We present a method of obtaining an equation of a minimal timelike surface in terms of canonical parameters, which play a role similar to the role of the natural parameters of curves in \(\mathbb R^3\). Having one holomorphic function that generates a minimal timelike surface, we find all holomorphic functions that generate the same surface. In this way we give a correspondence between a minimal timelike surface and a class of holomorphic functions. As an application, we prove that the Enneper surfaces are the only minimal timelike surfaces in \(\mathbb R^3_1\) with polynomial parametrization of degree 3 in isothermal parameters.

2020 Mathematics Subject Classification:

53A10, 53B30, 53C50

Keywords

timelike surfaces, canonical parameters, Weierstrass formula

Full text

Author Details

Ognian Kassabov

Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., Bl. 8
1113, Sofia, Bulgaria
e-mail: okassabov@math.bas.bg

Velichka Milousheva

Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., Bl. 8
1113, Sofia, Bulgaria
e-mail: vmil@math.bas.bg

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