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arXiv · 1610.07317

The geometry of stable minimal surfaces in metric Lie groups

William H. Meeks III·Pablo Mira·Joaquin Perez
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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.

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William H. Meeks III, Pablo Mira, Joaquin Perez. 2016-10-24. The geometry of stable minimal surfaces in metric Lie groups. https://arxiv.org/abs/1610.07317

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