Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space

2019/07/01 by Shintaro Akamine, Akamine, Shintaro, Masaaki Umehara +3 · 2 citations
Mathematics · #35M10 #53A10 #53B30 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1907.00739

openalex publication_date 2019/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Consider a surface S immersed in the Lorentz-Minkowski 3-space \boldsymbol R³₁. A complete light-like line in \boldsymbol R³₁ is called an entire null line on the surface S in \boldsymbol R³₁ if it lies on S and consists of only 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 \boldsymbol R², then it must be a light-like plane. Our example is critical in the sense that it is defined on a certain non-convex domain.

Citations

• Surfaces with light-like points in Lorentz-Minkowski 3-space with applications
• Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

Cited by

• Prescribing the mean curvature of an achronal hypersurface as a measure: the case of 3D spacetimes
• Examples of entire zero-mean curvature graphs of mixed-type in Lorentz-Minkowski space via Konderak's formulas

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