Chen, Bang-Yen Nonlinear Klein-Gordon equations and Lorentzian minimal surfaces in Lorentzian complex space forms. (English) Zbl 1172.53033 Taiwanese J. Math. 13, No. 1, 1-24 (2009). 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. Cited in 5 Documents 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 Keywords:Lorentzian surfaces; slant surfaces; minimal surfaces; nonlinear Klein-Gordon equation; Lorentzian complex space form; Lagrangian surface; Lagrangian surfaces of Klein-Gordon type PDFBibTeX XMLCite \textit{B.-Y. Chen}, Taiwanese J. Math. 13, No. 1, 1--24 (2009; Zbl 1172.53033) Full Text: DOI