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 | References | Similar Articles | Additional Information

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)