A Moser–Bernstein problem for Riemannian warped products

Abstract

In this work we deal with an elliptic non-linear problem, which arises naturally from Riemannian geometry. This problem has classically been studied in the the Euclidean n-dimensional space and it is known as the Moser–Bernstein problem. Nevertheless we solve this type of problems in a wide family of Riemannian manifolds, constructed as Riemannian warped products. More precisely, we study the entire solutions to the minimal hypersurface equation in a Riemannian warped product $$M=P\times _h{\mathbb {R}}$$, where P is a complete Riemannian parabolic manifold and h a positive smooth function on P.