LCN-translation surfaces in affine 3-space

Abstract

In this paper, we give the classification of the LCN-translation surfaces
with zero mean curvature and zero Gaussian curvature in 3-dimensional
Affine space.

Keywords

• Translation surfaces,affine 3-space,LCN- translation surfaces,LN- translation surfaces.

References

1. Andrade, M. Calculus of Affine Structures and Applications for Isosurfaces (in Portuguese), PhD dissertation, Rio de Janeiro, August 2011.
2. Andrade, M. and Lewiner, T. Affine-invariant Curvature Estimators for Implicit Surfaces, Computer Aided Geometric Design 29 (2012),162--173.
3. Arslan, K., Bayram, B., Bulca, B. and Ozturk, G. On translation surfaces in 4-dimensional Euclidean space, Acta Et Commentationes Universitatis Tartuensis De Mathematica, 20(2) (2016),123-133.
4. Blaschke, W. Vorlesungen über Differentialgeometrie II, Springer, Berlin, 1923.L. Verstraelen, L. Vrancken, Affine variation formulas and affine minimal surfaces, Michigan Math. J., 36 (1989), 77--93.
5. Bulca, B. On generalized LN-Surfaces in E⁴, Mathematical Sciences And Applications E-Notes, 1(2) (2013), 35-41 .
6. Goemans, W. Surfaces in three-dimensional Euclidean and Minkowski space, in particular a study of Weingarten surfaces, PhD. Dissertation, September 2010.
7. Huamani, E. F. C. Affine Minimal Surfaces with Singularities, Masters dissertation,Rio de Janeiro,September 2017.
8. Fu, Y. and Hou, Z.H. Affine Translation Surfaces with Constant Gauss Curvatures, Kyungpook Math. J., 50 (2010), 337-343.

9. Juttler, B. and Sampoli, M.L. Hermite interpolation by piecewise polynomial surfaces with rational offsets, Comp. Aided Geom. Design, 17 (2000), 361-385.
10. Magid, M. and Vrancken, L. Affine Translation Surfaces, Results Math., 35 (1999),134-144.
11. Manhart, F. Die Affin minimal rückungsflächen, Arch. Math., 44 (1985), 547-556.
12. Liu, H.L. Translation Surfaces with Constant Mean Curvature in 3-dimensional spaces, J.Geom., 64 (1999), 141-149.
13. Sampoli, M.L. Computing the convolution and the Minkowski sum of surfaces, Proceedings of the 21st Spring Conference on Computer Graphics, Budmerice,Slovakia, May 12-14, (2005),111-117
14. Sampoli, M. L., Peternell, M. and Jüttler, B. Rational surfaces with linear normals and their convolutions with rational surfaces, Comp. Aided Geom. Design, 23 (2006), 179-192 Sun, H. On affine translation surfaces of constant mean curvature, Kumamoto J. Math., 13 (2000) 49---57.
15. Yang, Y., Yu, Y.H. and Liu, H.L. Linear Weingarten centroaffine translation surfaces in R³, J. Math. Anal. Appl., 375 (2011), 458-466.
16. Yanga, D.Fu, Y.,On affine translation surfaces in affine space, J. Math. Anal. Appl., 440 (2016), 437-450. Yoon, D.W. Some Classification of Translation surfaces in Galilean 3-space, Int. Journal of Math. Analysis, 6(28) (2012), 1355-1361.