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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The geometry of stable minimal surfaces in metric Lie groups

Authors: William H. Meeks III, Pablo Mira and Joaquín Pérez
Journal: Trans. Amer. Math. Soc. 372 (2019), 1023-1056
MSC (2010): Primary 53A10; Secondary 49Q05, 53C42
Published electronically: February 22, 2019
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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 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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Additional Information

William H. Meeks III
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003

Pablo Mira
Affiliation: Department of Applied Mathematics and Statistics, Universidad Politécnica de Cartagena, E-30203 Cartagena, Murcia, Spain

Joaquín Pérez
Affiliation: Department of Geometry and Topology and Institute of Mathematics IEMath-GR, University of Granada, 18001 Granada, Spain

Keywords: Minimal surface, radius estimates, stability, Rad\'o's theorem, metric Lie group, homogeneous 3-manifold, left invariant metric
Received by editor(s): October 23, 2016
Received by editor(s) in revised form: October 24, 2016, and April 14, 2018
Published electronically: February 22, 2019
Additional Notes: This material is based upon work for the NSF under Award No. DMS-1309236. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF
The second author’s research was partially supported by MINECO-FEDER grant no. MTM2016-80313-P and Programa de Apoyo a la Investigación, Fundación Séneca-Agencia de Ciencia y Tecnología Región de Murcia, reference 19461/PI/14
The third author’s research was partially supported by MINECO/FEDER grants no. MTM2014-52368-P and MTM2017-89677-P
Article copyright: © Copyright 2019 American Mathematical Society