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

<p>Calabi’s Bernstein-type theorem asserts that <italic>a zero mean curvature entire graph in Lorentz-Minkowski space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper L cubed"> <mml:semantics> <mml:msup> <mml:mi mathvariant="bold-italic">L</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\boldsymbol {L}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which admits only space-like points is a space-like plane</italic>. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper E cubed"> <mml:semantics> <mml:msup> <mml:mi mathvariant="bold-italic">E</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\boldsymbol {E}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and maximal surfaces in Lorentz-Minkowski space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper L cubed"> <mml:semantics> <mml:msup> <mml:mi mathvariant="bold-italic">L</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\boldsymbol {L}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we give an improvement of this Bernstein-type theorem. More precisely, we show that <italic>a zero mean curvature entire graph in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper L cubed"> <mml:semantics> <mml:msup> <mml:mi mathvariant="bold-italic">L</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\boldsymbol {L}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which does not admit time-like points <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis"> <mml:semantics> <mml:mo stretchy="false">(</mml:mo> <mml:annotation encoding="application/x-tex">(</mml:annotation> </mml:semantics> </mml:math> </inline-formula>namely, a graph consists of only space-like and light-like points<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="right-parenthesis"> <mml:semantics> <mml:mo stretchy="false">)</mml:mo> <mml:annotation encoding="application/x-tex">)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a plane</italic>.</p>