Reflection Principles for Zero Mean Curvature Surfaces in the Simply Isotropic 3-space

Results in Mathematics - Tập 77 - Trang 1-13 - 2022
Shintaro Akamine1, Hiroki Fujino2
1Department of Liberal Arts, College of Bioresource Sciences, Nihon University, Fujisawa, Japan
2Institute for Advanced Research, Graduate School of Mathematics, Nagoya University, Nagoya, Japan

Tóm tắt

Zero mean curvature surfaces in the simply isotropic 3-space $${\mathbb {I}}^3$$ naturally appear as intermediate geometry between geometry of minimal surfaces in $${\mathbb {E}}^3$$ and that of maximal surfaces in $${\mathbb {L}}^3$$ . In this paper, we investigate reflection principles for zero mean curvature surfaces in $${\mathbb {I}}^3$$ as with the above surfaces in $${\mathbb {E}}^3$$ and $${\mathbb {L}}^3$$ . In particular, we show a reflection principle for isotropic line segments on such zero mean curvature surfaces in $${\mathbb {I}}^3$$ , along which the induced metrics become singular.

Tài liệu tham khảo

Ahlfors, L.V.: Complex Analysis, 3rd edn. International Series in Pure and Applied Mathematics, p. 331. McGraw-Hill Book Co., New York, (1978). An introduction to the theory of analytic functions of one complex variable Akamine, S., Fujino, H.: Reflection principle for lightlike line segments on maximal surfaces. Ann. Global Anal. Geom. 59(1), 93–108 (2021). https://doi.org/10.1007/s10455-020-09743-4 Akamine, S., Fujino, H.: Duality of boundary value problems for minimal and maximal surfaces. To appear in Comm. Anal. Geom. arXiv:1909.00975 Akamine, S., Fujino, H.: Extension of Krust theorem and deformations of minimal surfaces. Ann. Mat. Pura Appl. (4) https://doi.org/10.1007/s10231-022-01211-z. da Silva, L.C.B.: Holomorphic representation of minimal surfaces in simply isotropic space. J. Geom. 112(3), 35–21 (2021). https://doi.org/10.1007/s00022-021-00598-z Dierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal Surfaces, 2nd edn. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 339, p. 688. Springer. With assistance and contributions by A. Küster and R. Jakob. (2010). https://doi.org/10.1007/978-3-642-11698-8 Katznelson, Y.: An Introduction to Harmonic Analysis. In: Cambridge Mathematical Library, 3rd edn., p. 314. Cambridge University Press, Cambridge (2004). https://doi.org/10.1017/CBO9781139165372 Ma, X., Wang, C., Wang, P.: Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space. Adv. Math. 249, 311–347 (2013). https://doi.org/10.1016/j.aim.2013.09.013 Milnor, J.: Dynamics in One Complex Variable. In: Annals of Mathematics Studies, vol. 160, 3rd edn., p. 304. Princeton University Press, Princeton, NJ (2006) Pember, M.: Weierstrass-type representations. Geom. Dedicata. 204, 299–309 (2020). https://doi.org/10.1007/s10711-019-00456-y Pottmann, H., Grohs, P., Mitra, N.J.: Laguerre minimal surfaces, isotropic geometry and linear elasticity. Adv. Comput. Math. 31(4), 391–419 (2009). https://doi.org/10.1007/s10444-008-9076-5 Sachs, H.: Isotrope Geometrie des Raumes, p. 323. Friedr. Vieweg & Sohn, Braunschweig (1990). https://doi.org/10.1007/978-3-322-83785-1 Sato, Y.: \(d\)-minimal surfaces in three-dimensional singular semi-Euclidean space \({\mathbb{R} }^{0,2,1}\). Tamkang J. Math. 52(1), 37–67 (2021). https://doi.org/10.5556/j.tkjm.52.2021.3045 Seo, J.J., Yang, S.-D.: Zero mean curvature surfaces in isotropic three-space. Bull. Korean Math. Soc. 58(1), 1–20 (2021). https://doi.org/10.4134/BKMS.b190783 Strubecker, K.: Differentialgeometrie des isotropen Raumes. III. Flächentheorie. Math. Z. 48, 369–427 (1942). https://doi.org/10.1007/BF01180022 Strubecker, K.: Über Potentialflächen. Arch. Math. 5, 32–38 (1954). https://doi.org/10.1007/BF01899315 Strubecker, K.: Duale Minimalflächen des isotropen Raumes. Rad Jugoslav. Akad. Znan. Umjet. 382, 91–107 (1978)