Ando, Naoya Surfaces in pseudo-Riemannian space forms with zero mean curvature vector. (English) Zbl 1479.53021 Kodai Math. J. 43, No. 1, 193-219 (2020). This work characterizes a space-like surfaces in a pseudo-Riemannian space form with zero mean curvature vector, in terms of complex quadratic differentials on the surface as sections of a holomorphic line bundle sections of a holomorphic line bundle, and combining them, a holomorphic quartic differential.If the ambient space is \(S^4\) or \(S^4_1,\) then the differentials are given in some papers of R. Bryant and, in the second case, it coincides with a holomorphic quartic differential on a Willmore surface in \(S^3\) corresponding to the original surface through the conformal Gauss map.In addition, the author defines the conformal Gauss maps of surfaces in \(E^3\) and \(H^ 3\), and space-like surfaces in \(S^3_1,\) \(E^3_1,\) \( H^3_1\) and the cone of future-directed light-like vectors of \(E^4_1\), and obtains results which are analogous to those for the conformal Gauss map of a surface in \(S^3.\) Reviewer: Esther Sanabria (Valencia) Cited in 5 Documents MSC: 53B25 Local submanifolds 35N10 Overdetermined systems of PDEs with variable coefficients 49Q05 Minimal surfaces and optimization 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics Keywords:Willmore surface; conformal Gauss map; holomorphic quartic differential; complex quadratic differential; pseudo-Riemannian space form; zero mean curvature vector; space-like surface PDFBibTeX XMLCite \textit{N. Ando}, Kodai Math. J. 43, No. 1, 193--219 (2020; Zbl 1479.53021) Full Text: DOI Euclid