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

Abstract

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

9 pages, 2 figures

References in corpus (1)

• Surfaces with light-like points in Lorentz-Minkowski 3-space with applications

Cited by in corpus (2)

• Bernstein-type theorem for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski space
• Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space