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# Jenkins-Serrin problem for translating graphs

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.

# Asymptotic Dirichlet problems for the mean curvature operator

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.

Seminario 2ª Planta, IEMATH