Abstract
By solving certain (partial) differential equations we give the total classification of Weingarten affine translation surfaces in three dimensional Euclidean space \(\mathbb {E}^3\). Explicitly, a Weingarten affine translation surface in Euclidean 3-space is the minimal affine Scherk surface or the surface with flat metric.
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van-Brunt, B., Grant, K.: Potential applications of Weingarten surfaces in CAGD, part I: Weingarten surfaces and surface shape investigation. Comput. Aided Geom. Des. 13, 569–582 (1996)
Dillen, F., Goemans, W., Van de Woestyne, I.: Translation surfaces of Weingarten type in 3-space. Bulletin of the Transilvania University of Brasov, Vol. 1(50) (2008), Series III: Mathematics, Informatics, Physics, pp. 109–122
Kenmotsu, K.: Surfaces with Constant Mean Curvature. American Math. Soc., Translations of Math. Monographs, vol. 221 (2003)
Kühnel, W., Steller, M.: On closed Weingarten surfaces. Monatshefte für Mathematik 146, 113–126 (2005)
Li, A.M., Simon, U., Zhao, G.: Global Affine Differential Geometry of Hypersurfaces. W. De Gruyter, Berlin (1993)
Lie, S.: Gesammelte Abhandlungen, B. G. Teubner, Leibzig and H. Aschehoug & Co. Oslo (1934-1960)
Liu, H., Jung, S.D.: Affine translation surfaces with constant mean curvature in Euclidean 3-space. J. Geom. (2016). doi:10.1007/s00022-016-0348-9
Liu, H., Yu, Y.: Affine translation surfaces in Euclidean 3-space. Proc. Jpn. Acad. Math. Sci. 89(Ser. A), 111–113 (2013)
Nitsche, J.C.C.: Lecture on Minimal Surfaces, vol. 1. Cambridge University Press, Cambridge (1989)
Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the Affine Differential Geometry of Hypersurfaces. Lecture Notes, Science University Tokyo, ISBN 3-7983-1529-9 (1991)
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H. Liu was supported by NSFC (No. 11371080); Joint Research of NSFC and NRF; partially supported by the Chern Institute of Mathematics and Northeastern University. S. D. Jung was supported by NRF-2015R1A2A2A01003491.
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Jung, S.D., Liu, H. & Liu, Y. Weingarten Affine Translation Surfaces in Euclidean 3-Space. Results Math 72, 1839–1848 (2017). https://doi.org/10.1007/s00025-017-0737-x
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DOI: https://doi.org/10.1007/s00025-017-0737-x