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Hypersurfaces with light-like points in a Lorentzian manifold. (English) Zbl 1430.53009

Authors’ abstract: Consider a constant mean curvature immersion \(F:U(\subset \mathbb{R}^n)\rightarrow M\) into an arbitrary Lorentzian \((n+1)\)-manifold \(M\). A point \(o\in U\) is called a light-like point if the first fundamental form \(\text{d}s^2\) of \(F\) degenerates at \(o\). We denote by \(B_F\) the determinant function of the symmetric matrix associated to \(\text{d}s^2\) with respect to a local coordinate system at \(o\). A light-like point \(o\) is said to be degenerate if the exterior derivative of \(B_F\) vanishes at \(o\). We show that if \(o\) is a degenerate light-like point, then the image of \(F\) contains a light-like geodesic segment of \(M\) passing through \(f(o)\) (cf. Theorem E). This explains why several known examples of constant mean curvature hypersurface in the Lorentz-Minkowski \((n+1)\)-space form \(\mathbb{R}^{n+1}_1\) contain light-like lines on their sets of light-like points, under a suitable regularity condition of \(F\). Several related results are also given.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
35M10 PDEs of mixed type
53A35 Non-Euclidean differential geometry
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