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

Akamine, S.; Umehara, M.; Yamada, K.

doi:10.1090/bproc/44

Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality
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by S. Akamine, M. Umehara and K. Yamada HTML | PDF

Proc. Amer. Math. Soc. Ser. B 7 (2020), 17-27

Abstract:

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

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