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Affine translation surfaces with constant mean curvature in Euclidean 3-space. (English) Zbl 1380.53011

Summary: Using classical methods and solving certain differential equations we classify a kind of new surface, affine translation surfaces, which has the non-zero constant mean curvature in three dimensional Euclidean space \(\mathbf E^3\). Therefore a kind of solutions for the constant mean curvature equation is given.

MSC:

53A05 Surfaces in Euclidean and related spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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References:

[1] Kenmotsu, K.: Surfaces with Constant Mean Curvature, Translations of Math. Monographs, vol. 221. American Mathematical Society, Providence (2003) · Zbl 1042.53001
[2] Kenmotsu K.: Surfaces of revolution with periodic mean curvature. Osaka J. Math. 40(3), 687-696 (2003) · Zbl 1041.53005
[3] Liu, H., Yu, Y.: Affine translation surfaces in Euclidean 3-space. In: Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, vol. 89, pp. 111-113, Ser. A (2013) · Zbl 1287.53004
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