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Surfaces with light-like points in Lorentz-Minkowski 3-space with applications. (English) Zbl 1402.53007

Cañadas-Pinedo, María A. (ed.) et al., Lorentzian geometry and related topics. GeLoMa 2016, Málaga, Spain, September 20–23, 2016. Cham: Springer (ISBN 978-3-319-66289-3/hbk; 978-3-319-66290-9/ebook). Springer Proceedings in Mathematics & Statistics 211, 253-273 (2017).
Summary: With several concrete examples of zero mean curvature surfaces in the Lorentz-Minkowski 3-space \(\boldsymbol{R}^3_1\) containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the first fundamental form of a surface degenerates is said to be light-like. We also show a theorem on a property of light-like points of a surface in \(\boldsymbol{R}^3_1\) whose mean curvature vector is smoothly extendable. This explains why such surfaces will contain a light-like line when they do not change causal types. Moreover, several applications of these two results are given.
For the entire collection see [Zbl 1394.53001].

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
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