New results on the geometry of translation surfaces. (English) Zbl 1204.53005

Munteanu, Marian I.; Nistor, Ana I.

J. Geom. Symmetry Phys. 18, 49-62 (2010).

Summary: We study the second mean curvature for different hypersurfaces in space forms. We furnish some examples and we recall some connections between minimality and biharmonicity. The main result consists in proving that there are no minimal translation surfaces in the Euclidean three-space.

MSC:

53A05	Surfaces in Euclidean and related spaces
53B25	Local submanifolds
53C40	Global submanifolds
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

second fundamental form; second Gaussian curvature; second mean curvature