I will discuss about recent joint works with E.S. Gama, J. de Lira and F. Martín concerning Jenkins-Serrin type problems for the graphical translators of the mean curvature flow in Riemannian product manifolds \(M\times R\). We prove, for example, the existence of Jenkins-Serrin type translators that can be described as horizontal graphs "over" suitable domains in a vertical plane.

In \(R^n\) (\(n\) 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 \(R^n\) are constant in every dimension by a result due to Bombieri, De Giorgi and Miranda. If the underlying space is changed from \(R^n\) to a negatively curved Riemannian manifold, the situation is completely different. Namely, if the sectional curvature of \(M\) satisfies suitable bounds, then \(M\) 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 f-minimal graphs. The talk is based on joint works with Jean-Baptiste Casteras and Ilkka Holopainen.