# Geometry Seminar

#### Event Details

Title: Stable minimal surfaces in semidirect products

Abstract: We will 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 an priori radius estimate which depends only on the maximum distance of points of the boundary $\partial M$ to a vertical geodesic of $X$. In particular, there are no complete stable minimal surfaces inside solid metric cylinders around vertical geodesics in $X$.​ We also give a generalization of the classical Rado's Theorem 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.​ Joint work with Bill Meeks and Pablo Mira.​