The geometry of stable minimal surfaces in metric Lie groups

Meeks, William, III; Mira, Pablo; Pérez, Joaquín

doi:10.1090/tran/7634

The geometry of stable minimal surfaces in metric Lie groups
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by William H. Meeks III, Pablo Mira and Joaquín Pérez PDF

Trans. Amer. Math. Soc. 372 (2019), 1023-1056 Request permission

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 an 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 Radó 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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