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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 \(\mathbb C_1^2\). Finally, we classify minimal slant surfaces in the Lorentzian complex projective plane \(\mathbb CP_1^2(4)\)and in the Lorentzian complex hyperbolic plane \(\mathbb CH_1^2(-4)\). In particular, our latter results show that if a minimal slant surface in \(\mathbb CP_1^2(4)\) or in \(\mathbb CH_1^2(-4)\) 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
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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