Resumen
After a brief look into classics problems in minimal surfaces theory, such as Plateau problem, we will get into graph theory.
We will focus in the study of the Dirichlet problem, distinguishing bounded domains whose boundary values are either finite or include infinite values as well. When it comes to the latter case we will prove a remarkable theorem by Jenkins and Serrin. While in the original proof classics techniques of minimal surfaces theory are used, in this work we will make use of newer tools related to divergence lines and flux in order to go get the same point as in Jenkins-Serrin.
We will work all the time with any space $\mathbb{M} \times \mathbb{R}$ with $\mathbb{M}$ a 2-Riemannian variety, although specifying the results for $\mathbb{R}^3$ and $\mathbb{H}^2\times \mathbb{R}$, product space of which we will study some of the main characteristics.