Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space. (English) Zbl 1441.53005

Summary: Consider a surface \(S\) immersed in the Lorentz-Minkowski 3-space \(\mathbb{R}^3_1\). A complete light-like line in \(\mathbb{R}^3_1\) is called an entire null line on \(S^3_1\) if it lies on \(S\) and consists only of null points with respect to the induced metric. In this paper, we show the existence of embedded space-like maximal graphs containing entire null lines. If such a graph is defined on a convex domain in \(\mathbb{R}^2 \), then it must be contained in a light-like plane (cf. Remark 3.3). Our example is critical in the sense that it is defined on a certain non-convex domain.


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
Full Text: DOI arXiv Euclid


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