Previous |  Up |  Next

Article

Keywords:
Timelike minimal surface; bicomplex numbers; timelike minimal surface with ends; complete maximal surface; zero mean curvature surface
Summary:
In this paper, we show the existence of a timelike minimal surface with an arbitrary number of weak complete ends. Then, we discuss the asymptotic behaviour of the simple ends and the topology of the singularity set of the constructed timelike minimal surface.
References:
[1] Akamine, Shintaro: Behavior of the Gaussian curvature of timelike minimal surfaces with singularities. Hokkaido Math. J. 48 (2019), no. 3, 537 – 568.
[2] Charak, Kuldeep Singh, Rochon, Dominic, Sharma, Narinder: Normal families of bicomplex meromorphic functions. Ann. Polon. Math. 103 (2012), no. 3, 303–317. DOI 10.4064/ap103-3-6
[3] Erdem, Sadettin: Harmonic maps of Lorentz surfaces, quadratic differentials and paraholomorphicity. Beitr. Algebra Geom. 38 (1997), no. 1, 19–32. MR 1447983
[4] Estudillo, Francisco J. M., Romero, Alfonso: Generalized maximal surfaces in Lorentz-Minkowski space $\mathbb{L}^3$. Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 515–524. DOI 10.1017/S0305004100075587
[5] Hashimoto, Kaname, Kato, Shin: Bicomplex extensions of zero mean curvature surfaces in $\mathbb{R}^{2,1}$ and $\mathbb{R}^{2,2}$. J. Geom. Phys. 138 (2019), 223–240. DOI 10.1016/j.geomphys.2018.12.017
[6] ichi Inoguchi, Jun, Toda, Mitsuhiro: Timelike Minimal Surfaces via Loop Groups. Acta Appl. Math. 83 (2004), 313–355. DOI 10.1023/B:ACAP.0000039015.45368.f6
[7] Imaizumi, Taishi: MAXIMAL SURFACES WITH SIMPLE ENDS. Kyushu J. Math. 58 (2004), no. 1, 59–70.
[8] Jorge, Luquesio P., Meeks, William H.: The topology of complete minimal surfaces of finite total Gaussian curvature. Topology 22 (1983), no. 2, 203–221. DOI 10.1016/0040-9383(83)90032-0
[9] Kim, Young Wook, Koh, Sung-Eun, Shin, Heayong, Yang, Seong-Deog: Spacelike maximal surfaces, timelike minimal surfaces, and Björling representation formulae. J. Korean Math. Soc. 48 (2011), no. 5, 1083–1100. MR 2850077 DOI 10.4134/JKMS.2011.48.5.1083
[10] Kobayashi, Osamu: Maximal surfaces in the $3$-dimensional Minkowski space $\mathbb{L}^3$. Tokyo J. Math. 6 (1983), no. 2, 297–309. MR 732085 DOI 10.3836/tjm/1270213872
[11] Konderak, Jerzy J.: A Weierstrass representation theorem for Lorentz surfaces. Complex Var. Theory Appl. 50 (2005), no. 5, 319–332.
[12] Kumar, Pradip, Mohanty, Sai Rasmi Ranjan: Genus zero complete maximal maps and maxfaces with an arbitrary number of ends. C. R. Math. Acad. Sci. Paris 361 (2023), 1683–1690. MR 4683343 DOI 10.5802/crmath.525
[13] Lin, Senchun, Weinstein, Tilla: $\mathbb{E}^{3}$-complete timelike surfaces in $\mathbb{E}^{3}_{1}$ are globally hyperbolic. Michigan Math. J. 44 (1997), no. 3, 529–541. MR 1481117
[14] Luna-Elizarrarás, M. Elena, Pérez-Regalado, C. Oscar, Shapiro, Michael: Singularities of bicomplex holomorphic functions. Math. Models Methods Appl. Sci. (2021), 1–16.
[15] Luna-Elizarrarás, M. Elena, Shapiro, Michael, Struppa, Daniele C., Vajiac, Adrian: Bicomplex Holomorphic Functions. 1 ed., Frontiers in Mathematics, Birkhäuser, Cham, 2015.
[16] McNertney, L.: One-parameter families of surfaces with constant curvature in Lorentz 3-space. Ph.d. thesis, Brown University, 1980.
[17] Umehara, Masaaki, Yamada, Kotaro: Maximal surfaces with singularities in Minkowski space. Hokkaido Math. J. 35 (2006), no. 1, 13–40. MR 2225080
[18] Weinstein, Tilla: An introduction to Lorentz surfaces. De Gruyter Expositions in Mathematics, vol. 22, Walter de Gruyter & Co., Berlin, 1996. MR 1405166
Partner of
EuDML logo