Surfaces immersed in $\mathbb{R}^3_+$ with the same Gauss curvature induced by the Euclidean and hyperbolic metrics

[mathjax] We will consider the geometric problem of findind immersed surfaces in the upper half-space such that the Gaussian curvatures induced from the ambient euclidean and hyperbolic metrics coincide. We will show the surprising geometric connection between such immersions and minimal surfaces (in euclidean space and timelike surfaces in Minkowski space). This connection enables the construction of infinitly many non trivial examples, we will discuss some of them in the talk.Joint work with Nilton Barroso.