×

Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality. (English) Zbl 1433.53017

Summary: 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.

MSC:

53A35 Non-Euclidean differential geometry
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35M10 PDEs of mixed type
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akamine, Shintaro, Causal characters of zero mean curvature surfaces of Riemann type in the Lorentz-Minkowski 3-space, Kyushu J. Math., 71, 2, 211-249 (2017) · Zbl 1409.53010
[2] Akamine, Shintaro; Singh, Rahul Kumar, Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born-Infeld equation, Proc. Indian Acad. Sci. Math. Sci., 129, 3, Art. 35, 18 pp. (2019) · Zbl 1417.53010
[3] Akamine, Shintaro; Umehara, Masaaki; Yamada, Kotaro, Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space, Proc. Japan Acad. Ser. A Math. Sci., 95, 9, 97-102 (2019) · Zbl 1441.53005
[4] Bers, Lipman, Mathematical aspects of subsonic and transonic gas dynamics, Surveys in Applied Mathematics, Vol. 3, xv+164 pp. (1958), John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London · Zbl 1348.82003
[5] Calabi, Eugenio, Examples of Bernstein problems for some nonlinear equations. Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), 223-230 (1970), Amer. Math. Soc., Providence, R.I.
[6] Cheng, Shiu Yuen; Yau, Shing Tung, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2), 104, 3, 407-419 (1976) · Zbl 0352.53021
[7] Ecker, Klaus, Area maximizing hypersurfaces in Minkowski space having an isolated singularity, Manuscripta Math., 56, 4, 375-397 (1986) · Zbl 0594.58023
[8] Fernandez, Isabel; Lopez, Francisco J., On the uniqueness of the helicoid and Enneper’s surface in the Lorentz-Minkowski space \(\mathbb{R}^3_1\), Trans. Amer. Math. Soc., 363, 9, 4603-4650 (2011) · Zbl 1245.53012
[9] Fujimori, Shoichi; Kawakami, Yu; Kokubu, Masatoshi; Rossman, Wayne; Umehara, Masaaki; Yamada, Kotaro, Entire zero-mean curvature graphs of mixed type in Lorentz-Minkowski 3-space, Q. J. Math., 67, 4, 801-837 (2016) · Zbl 1364.53012
[10] Fujimori, S.; Kim, Y. W.; Koh, S.-E.; Rossman, W.; Shin, H.; Takahashi, H.; Umehara, M.;  Yamada, K.; Yang, S.-D., Zero mean curvature surfaces in \(\mathbf{L}^3\) containing a light-like line, C. R. Math. Acad. Sci. Paris, 350, 21-22, 975-978 (2012) · Zbl 1257.53090
[11] Fujimori, S.; Kim, Y. W.; Koh, S.-E.; Rossman, W.; Shin, H.; Umehara, M.; Yamada, K.; Yang, S.-D., Zero mean curvature surfaces in Lorentz-Minkowski 3-space and 2-dimensional fluid mechanics, Math. J. Okayama Univ., 57, 173-200 (2015) · Zbl 1320.53017
[12] Gu, Chao Hao, The extremal surfaces in the \(3\)-dimensional Minkowski space, Acta Math. Sinica (N.S.), 1, 2, 173-180 (1985) · Zbl 0595.49027
[13] Hartman, Philip; Nirenberg, Louis, On spherical image maps whose Jacobians do not change sign, Amer. J. Math., 81, 901-920 (1959) · Zbl 0094.16303
[14] Hashimoto, Kaname; Kato, Shin, Bicomplex extensions of zero mean curvature surfaces in \(\mathbf{R}^{2,1}\) and \(\mathbf{R}^{2,2} \), J. Geom. Phys., 138, 223-240 (2019) · Zbl 1414.53052
[15] Hoffman, D.; Meeks, W. H., III, The strong halfspace theorem for minimal surfaces, Invent. Math., 101, 2, 373-377 (1990) · Zbl 0722.53054
[16] Kim, Young Wook; Koh, Sung-Eun; Shin, Heayong; Yang, Seong-Deog, Spacelike maximal surfaces, timelike minimal surfaces, and Bj\"{o}rling representation formulae, J. Korean Math. Soc., 48, 5, 1083-1100 (2011) · Zbl 1250.53016
[17] Klyachin, V. A., Surfaces of zero mean curvature of mixed type in Minkowski space, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Math., 67 67, 2, 209-224 (2003) · Zbl 1076.53015
[18] Kobayashi, Osamu, Maximal surfaces in the \(3\)-dimensional Minkowski space \(L^3 \), Tokyo J. Math., 6, 2, 297-309 (1983) · Zbl 0535.53052
[19] O’Neill, Barrett, Semi-Riemannian geometry, Pure and Applied Mathematics 103, xiii+468 pp. (1983), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York · Zbl 0531.53051
[20] Umehara, Masaaki; Yamada, Kotaro, Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J., 35, 1, 13-40 (2006) · Zbl 1109.53016
[21] Umehara, Masaaki; Yamada, Kotaro, Surfaces with light-like points in Lorentz-Minkowski 3-space with applications. Lorentzian geometry and related topics, Springer Proc. Math. Stat. 211, 253-273 (2017), Springer, Cham · Zbl 1402.53007
[22] Umehara, M.; Yamada, K., Hypersurfaces with light-like points in a Lorentzian manifold, J. Geom. Anal., 29, 4, 3405-3437 (2019) · Zbl 1430.53009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.