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Proceedings of the American Mathematical Society Series B

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


Authors: S. Akamine, M. Umehara and K. Yamada
Journal: Proc. Amer. Math. Soc. Ser. B 7 (2020), 17-27
MSC (2010): Primary 53A10; Secondary 35M10
DOI: https://doi.org/10.1090/bproc/44
Published electronically: February 20, 2020
MathSciNet review: 4066478
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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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Additional Information

S. Akamine
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
MR Author ID: 1232673
Email: s-akamine@math.nagoya-u.ac.jp

M. Umehara
Affiliation: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan
MR Author ID: 237419
Email: umehara@is.titech.ac.jp

K. Yamada
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
MR Author ID: 243885
Email: kotaro@math.titech.ac.jp

Keywords: Zero mean curvature, Lorentz-Minkowski space, Bernstein-type theorem, fluid mechanics, Chaplygin gas flow.
Received by editor(s): December 18, 2018
Received by editor(s) in revised form: June 25, 2019, and October 5, 2019
Published electronically: February 20, 2020
Additional Notes: The first author was supported in part by Grant-in-Aid for Young Scientists No. 19K14527 and for Scientific Research on Innovative Areas No. 17H06466
The second author was supported in part by Grant-in-Aid for Scientific Research (A) No. 26247005
The third author was supported in part by part by Grant-in-Aid for Scientific Research (B) No. 17H02839 from Japan Society for the Promotion of Science
All three authors were supported by JSPS/FWF Bilateral Joint Project I3809-N32 “Geometric Shape Generation”
Communicated by: Jiaping Wang
Article copyright: © Copyright 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)