Some fundamental theorems for isometric immersions into three-dimensional unimodular metric Lie groups

We will discuss the fundamental equations satisfied by an isometric immersion of a Riemannian surface into a three-dimensional unimodular Lie group endowed with a left-invariant Riemannian metric. This allows us to address two problems: first, to what extent the left-invariant Gauss map determines the isometric immersion, and second, under which assumptions a Lawson-type correspondence exists for constant mean curvature surfaces in unimodular metric Lie groups. We will also comment on how some of these results extend to the semi-Riemannian case