The geometry of stable minimal surfaces in metric Lie groups

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of ${\mathbb{R}}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depends only on the maximum distance of points of the boundary $\mathrm{\partial }M$ to a vertical geodesic of $X$. We also give a generalization of the classical Rado’s Theorem in ${\mathbb{R}}^3$ to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in $X$, and we study the geometry, existence and uniqueness of this type of Plateau problem.