Umehara, Masaaki; Yamada, Kotaro Hypersurfaces with light-like points in a Lorentzian manifold. II. (English) Zbl 1469.53016 Kodai Math. J. 44, No. 1, 69-76 (2021). Summary: In the authors’ previous work [J. Geom. Anal. 29, No. 4, 3405–3437 (2019; Zbl 1430.53009)], it was shown that if a zero mean curvature \(C^4\)-differentiable hypersurface in an arbitrarily given Lorentzian manifold admits a degenerate light-like point, then the hypersurface contains a light-like geodesic segment passing through the point. The purpose of this paper is to point out that the same conclusion holds with just \(C^3\)-differentiability of the hypersurfaces. MSC: 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35M10 PDEs of mixed type 53B30 Local differential geometry of Lorentz metrics, indefinite metrics Keywords:light-like geodesic; Lorentzian manifold; maximal hypersurface; zero mean curvature Citations:Zbl 1430.53009 PDFBibTeX XMLCite \textit{M. Umehara} and \textit{K. Yamada}, Kodai Math. J. 44, No. 1, 69--76 (2021; Zbl 1469.53016) Full Text: DOI arXiv References: [1] S. Akamine, A. Honda, M. Umehara and K. Yamada, Bernstein-type theorem for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski space, Bull. Braz. Math. Soc. (N.S.) (2020), DOI 10.1007/s00574-020-00196-8. · Zbl 1461.53005 [2] V. A. Klyachin, Zero mean curvature surfaces of mixed type in Minkowski space, Izv. Math. 67 (2003), 209-224. · Zbl 1076.53015 [3] M. Umehara and K. Yamada, Hypersurfaces with light-like points in a Lorentzian manifold, J. Geom. Anal. 29 (2019), 3405-3437. · Zbl 1430.53009 [4] M. Umehara and K. Yamada, Surfaces with light-like points in Lorentz-Minkowski 3-space with applications, Lorentzian geometry and related topics, Springer Proceedings in Mathematics & Statistics 211, Springer-Verlag, 2017, 253-273. · Zbl 1402.53007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.