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Nonlinear Klein-Gordon equations and Lorentzian minimal surfaces in Lorentzian complex space forms. (English) Zbl 1172.53033
Summary: We investigate Lorentzian minimal surfaces in Lorentzian complex space forms. First, we prove that for such surfaces the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Next, we classify Lorentzian minimal surfaces in the Lorentzian complex plane ${ℂ}_{1}^{2}$. Finally, we classify minimal slant surfaces in the Lorentzian complex projective plane $ℂ{P}_{1}^{2}\left(4\right)$and in the Lorentzian complex hyperbolic plane $ℂ{H}_{1}^{2}\left(-4\right)$. In particular, our latter results show that if a minimal slant surface in $ℂ{P}_{1}^{2}\left(4\right)$ or in $ℂ{H}_{1}^{2}\left(-4\right)$ contains no open subset of constant curvature, then it is of Klein-Gordon type which arises from the solutions of certain nonlinear Klein-Gordon equations.
##### MSC:
 53C40 Global submanifolds (differential geometry) 53C42 Immersions (differential geometry) 53C50 Lorentz manifolds, manifolds with indefinite metrics