Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

2019/04/17 by Shintaro Akamine, Akamine, Shintaro, Masaaki Umehara +3 · 3 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1904.08046

Abstract

Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space \boldsymbol L³ which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space \boldsymbol E³ and maximal surfaces in Lorentz-Minkowski space \boldsymbol L³, we give an improvement of this Bernstein-type theorem. More precisely, we show that a zero mean curvature entire graph in \boldsymbol L³ which does not admit time-like points (namely, a graph consists of only space-like and light-like points) is a plane.

Cited by

• Duality of boundary value problems for minimal and maximal surfaces
• Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space
• Examples of entire zero-mean curvature graphs of mixed-type in Lorentz-Minkowski space via Konderak's formulas

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• Zero mean curvature entire graphs of mixed type in Lorentz-Minkowski 3-space
• Examples of entire zero-mean curvature graphs of mixed-type in Lorentz-Minkowski space via Konderak's formulas
• Bernstein type theorems for spacelike stationary graphs in Minkowski spaces