Minimal Affine Translation Surfaces in Nil3 Space

doi:10.12068/j.issn.1005-3026.2024.10.018

Abstract

Abstract:

Nil3 space is a Riemannian space with a Heisenberg group structure. The affine translation surfaces in Nil₃ space are constructed by using the group operator of the Heisenberg group. These surfaces are generated by two planar curves as base lines through group operation. However， since group operations are not commutative， selecting the same base lines generates two types of different affine translation surfaces. Then， the two affine translation surfaces are classified. Employ the method of variation of constants and Euler’s method of indeterminate exponential functions to solve the differential equation arising when the mean curvature of a surface equal zero， and present the classification theorem for minimal affine translation surfaces under various operations. Finally， some specific minimal affine translation surfaces are given， and corresponding images are drawn using Mathematica.

Key words: Nil₃ space, affine translation surface, minimal surface, Heisenberg group, mean curvature

CLC Number:

O 186

Yan-hua YU, Qian-rong LI, Rui XUE. Minimal Affine Translation Surfaces in Nil₃ Space[J]. Journal of Northeastern University(Natural Science), 2024, 45(10): 1513-1520.

Figures/Tables 8

Fig.1 The first type of minimal affine translation surface M1 in case u″-a=0

Fig.2 The first type of minimal affine translation surface M1 in case B'=0,b1≠0

Fig.3 The first type of minimal affine translation surface M1 in case B'=0,b1=0

Fig.4 The first type of minimal affine translation surface M1 in case B'≠0,b6=0

Fig.5 The second type of minimal affine translation surface M2 in case u″=0

Fig.6 The second type of minimal affine translation surface M2 in case A1'=0,b=0

Fig.7 The second type of minimal affine translation surface M2 in case A1'=0,2ab=1

Fig.8 The second type of minimal affine translation

References 14

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[2]	QIAN Jin-hua， FU Xue-shan. Canal Surfaces in 3D Minkowski Space [J]. Journal of Northeastern University Natural Science, 2019, 40(1): 150-152.
[3]	YUAN Yuan， MA Longbiao， LIU Huili. Affine Translation Surfaces in Minkowski 3DSpace [J]. Journal of Northeastern University, 2013, 34(10): 1517-1519.

Minimal Affine Translation Surfaces in Nil₃ Space

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