The geometry of stable minimal surfaces in metric Lie groups

TL;DR

This paper investigates the geometric properties, radius estimates, and boundary value problems of stable minimal surfaces within certain homogeneous 3-manifolds modeled as semidirect products, extending classical results to these curved spaces.

Contribution

It provides a priori radius estimates based on boundary distances, generalizes Rado's Theorem to these manifolds, and explores existence and uniqueness of minimal surfaces with graphical boundaries.

Findings

A priori radius estimates depending on boundary distance

Generalization of Rado's Theorem in homogeneous 3-manifolds

Results on existence and uniqueness of Plateau problems in these spaces

Abstract

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 $\partial M$ 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.