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

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.