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Transactions of the American Mathematical Society

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

 
 

 

Minimal surfaces in minimally convex domains


Authors: Antonio Alarcón, Barbara Drinovec Drnovšek, Franc Forstnerič and Francisco J. López
Journal: Trans. Amer. Math. Soc. 371 (2019), 1735-1770
MSC (2010): Primary 53A10; Secondary 32B15, 32E30, 32H02
DOI: https://doi.org/10.1090/tran/7331
Published electronically: June 7, 2018
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Abstract: In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $ M$ into a minimally convex domain $ D\subset \mathbb{R}^3$ can be approximated uniformly on compacts in $ \mathring M=M\setminus bM$ by proper complete conformal minimal immersions $ \mathring M\to D$. We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in $ \mathbb{R}^3$, and we characterize the minimal surface hull of a compact set $ K$ in $ \mathbb{R}^n$ for any $ n\ge 3$ by sequences of conformal minimal discs whose boundaries converge to $ K$ in the measure theoretic sense.


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Additional Information

Antonio Alarcón
Affiliation: Departmento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, E–18071 Granada, Spain
Email: alarcon@ugr.es

Barbara Drinovec Drnovšek
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
Email: barbara.drinovec@fmf.uni-lj.si

Franc Forstnerič
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
Email: franc.forstneric@fmf.uni-lj.si

Francisco J. López
Affiliation: Departmento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, E–18071 Granada, Spain
Email: fjlopez@ugr.es

DOI: https://doi.org/10.1090/tran/7331
Keywords: Riemann surface, minimal surface, minimally convex domain
Received by editor(s): August 5, 2016
Received by editor(s) in revised form: June 27, 2017
Published electronically: June 7, 2018
Additional Notes: The first author was supported by the Ramón y Cajal program of the Spanish Ministry of Economy and Competitiveness.\endgraf The first and fourth authors were partially supported by the MINECO/FEDER grants MTM2014-52368 and MTM2017-89677-P, Spain.
The second and third authors were partially supported by the research program P1-0291 and grants J1-5432 and J1-7256 from ARRS, Republic of Slovenia.
Dedicated: Dedicated to Josip Globevnik for his seventieth birthday
Article copyright: © Copyright 2018 American Mathematical Society