A Bernstein-type theorem for Riemannian manifolds with a Killing field

Author: Alías Luis  

Publisher: Springer Publishing Company

ISSN: 0232-704X

Source: Annals of Global Analysis and Geometry, Vol.31, Iss.4, 2007-06, pp. : 363-373

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Abstract

The classical Bernstein theorem asserts that any complete minimal surface in Euclidean space </equationsource> that can be written as the graph of a function on </equationsource> 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.