Abstract
The classical Bernstein theorem asserts that any complete minimal surface in Euclidean space \(\mathbb{R}^3\) that can be written as the graph of a function on \(\mathbb{R}^2\) must be a plane. In this paper, we extend Bernstein’s result to complete minimal surfaces in (may be non-complete) ambient spaces of non-negative Ricci curvature carrying a Killing field. This is done under the assumption that the sign of the angle function between a global Gauss map and the Killing field remains unchanged along the surface. In fact, our main result only requires the presence of a homothetic Killing field.
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L.J. Alías was partially supported by MEC/FEDER project MTM2004-04934-C04-02, F. Séneca project 00625/PI/04, and F. Séneca grant 01798/EE/05, Spain
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Alías, L.J., Dajczer, M. & Ripoll, J. A Bernstein-type theorem for Riemannian manifolds with a Killing field. Ann Glob Anal Geom 31, 363–373 (2007). https://doi.org/10.1007/s10455-006-9045-5
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DOI: https://doi.org/10.1007/s10455-006-9045-5