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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 1 2 . Finally, we classify minimal slant surfaces in the Lorentzian complex projective plane P 1 2 (4)and in the Lorentzian complex hyperbolic plane H 1 2 (-4). In particular, our latter results show that if a minimal slant surface in P 1 2 (4) or in H 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:
53C40Global submanifolds (differential geometry)
53C42Immersions (differential geometry)
53C50Lorentz manifolds, manifolds with indefinite metrics