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 L 3 \boldsymbol {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 E 3 \boldsymbol {E}^3 and maximal surfaces in Lorentz-Minkowski space L 3 \boldsymbol {L}^3 , we give an improvement of this Bernstein-type theorem. More precisely, we show that a zero mean curvature entire graph in L 3 \boldsymbol {L}^3 which does not admit time-like points ( ( namely, a graph consists of only space-like and light-like points ) ) is a plane.