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Surfaces in pseudo-Riemannian space forms with zero mean curvature vector. (English) Zbl 07196516
This work characterize a space-like surface 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 differential are given in some papers of R. Bryant and, in the second case, 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, 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 have results which are analogous to those for the conformal Gauss map of a surface in \(S^3.\)

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
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