In RnRn (nn at most 7), the famous Bernstein's theorem states that every entire solution to the minimal graph equation must be affine. Moreover, entire positive solutions in RnRn are constant in every dimension by a result due to Bombieri, De Giorgi and Miranda. If the underlying space is changed from RnRn to a negatively curved Riemannian manifold, the situation is completely different. Namely, if the sectional curvature of MM satisfies suitable bounds, then MM possess a wealth of solutions. One way to study the existence of entire, continuous, bounded and non-constant solutions, is to solve the asymptotic Dirichlet problem on Cartan-Hadamard manifolds. In this talk I will discuss about recent existence results for minimal graphs and ff-minimal graphs. The talk is based on joint works with Jean-Baptiste Casteras and Ilkka Holopainen.
