$d$-Minimal Surfaces in Three-Dimensional Singular Semi-Euclidean Space $\mathbb{R}^{0,2,1}$

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Published: Jan 31, 2021
DOI: https://doi.org/10.5556/j.tkjm.52.2021.3045
Keywords:
minimal surface isotropic geometry semi-Riemannian geometry

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Yuichiro Sato

Abstract

In this paper, we investigate surfaces in singular semi-Euclidean space $\mathbb{R}^{0,2,1}$ endowed with a degenerate metric. We define $d$-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we prove that $d$-minimal surfaces in $\mathbb{R}^{0,2,1}$ and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space $\mathbb{R}^{4}_{1}$ are in one-to-one correspondence.

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How to Cite
Sato, Y. (2021). $d$-Minimal Surfaces in Three-Dimensional Singular Semi-Euclidean Space $\mathbb{R}^{0,2,1}$. Tamkang Journal of Mathematics, 52(1), 37–67. https://doi.org/10.5556/j.tkjm.52.2021.3045
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Vol. 52 No. 1 (2021)
Section
Papers
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

References

L. J. Al ́as, B. Palmer, Curvature properties of zero mean curvature surfaces in four- dimensional Lorentzian space forms, Math. Proc. Camb. Phil. Soc. 124 (1998), 315–327.

H. Anciaux, Minimal submanifolds in pseudo-Riemannian geometry, World Scientific (2011).

M. E. Aydin, Classification results on surfaces in the isotropic 3-space, AKU J. Sci. Eng. 16 (2016), 239–246.

M. E. Aydin, Constant curvature surfaces in a pseudo-isotropic space,Tamkang J. Math. 49 (2018), 221–233.

A. Bejancu and K. L. Duggal, Lightlike submanifolds of semi-Riemannian manifolds and applications, Kluwer Academic Publishers (1996).

B. Y. Chen, Black holes, marginally trapped surfaces and quasi-minimal surfaces, Tamkang J. Math. 40 (2009), 313–341.

Z. Erjavec, B. Divjak and D. Horvat, The general solutions of Frenet’s system in the equiform geometry of the Galilean, pseudo-Galilean, simple isotropic and double isotropic space, Int. Math. Forum 6 (2011), no. 17, 837–856.

F. J. M. Estudillo and A. Romero, On maximal surfaces in the n-dimensional Lorentz- Minkowski space, Geom. Dedicata 38 (1991), 167–174.

S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara and K. Yamada, En- tire zero-mean curvature graphs of mixed type in Lorentz-Minkowski 3-space, Quarterly J. Math. 67 (2016), 801–837.

S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces, Math. Z. 259 (2008), 827–848.

T. Ishihara, Maximal space like submanifolds of a pseudo-Riemannian space of constant curvature, Mich. Math. J. 35 (1988), 345–352.

X. Ma, C. P. Wang and P. Wang, Global geometry and topology of spacelike stationary sur- faces in the 4-dimensional Lorentz space, Adv. Math. 249 (2013), 311–347.

K. Nomizu and T. Sasaki, Affine differential geometry, Cambridge University Press (1994).

M. O’Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, London (1983).

A. L. Onishchik and R. Sulanke, Projective and Cayley-Klein geometries, Springer (2006).

H. Pottmann, P. Grohs and N. J. Mitra, Laguerre minimal surfaces, isotropic geometry and linear elasticity, Adv. Comput. Math. 31 (2009), 391–419.

H. Sachs, Isotrope Geometrie des Raumes, Vieweg, Braunschewig/Wiesbaden(1990).

K. Saji, Criteria for D4 singularities of wavefronts, Tohoku Math. J. 63 no.1 (2011), 137–147.

Y. Sato, On the classification of ruled minimal surfaces in pseudo-Euclidean space, Math. J. Okayama Univ. 61 (2019), 173–186.

L. C. B. da Silva, Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces, To appear in Tamkang J. Math.

L. C. B. da Silva, The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces, J. Geom. 110: 31 (2019).

L. C. B. da Silva, Differential geometry of invariant surfaces in simply and pseudo isotropic spaces, To appear in Math. J. Okayama Univ.

O. C. Stoica, On singular semi-Riemannian manifolds, Int. J. Geom. Methods Mod. Phys. 11, (2014), no. 5, 1450041.

M. Umehara and K. Yamada, Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J. 35 (2006), 13–40.

W. O. Vogel, U ber lineare Zusammenha ̈nge in singula ̈ren Riemannschen Ra ̈umen, Archiv der Mathematik 16 (1965), 106–116.