Abstract
In this paper, we prove that every conformal minimal immersion of an open Riemann surface into \({\mathbb {R}}^n\) for \(n\ge 5\) can be approximated uniformly on compacts by conformal minimal embeddings (see Theorem 1.1). Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into \({\mathbb {R}}^5\) (see Theorem 1.2). One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to \({\mathbb {R}}^n\) for any \(n\ge 3\) which is also proved in the paper (see Theorem 5.3).
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Acknowledgments
A. Alarcón is supported by the Ramón y Cajal program of the Spanish Ministry of Economy and Competitiveness. A. Alarcón and F. J. López are partially supported by the MINECO/FEDER Grants MTM2011-22547 and MTM2014-52368-P, Spain. F. Forstnerič is partially supported by the research program P1-0291 and the Grant J1-5432 from ARRS, Republic of Slovenia. Part of this work was made when F. Forstnerič visited the institute IEMath-Granada with support by the GENIL-SSV 2014 program. We wish to thank an anonymous referee for the remarks which lead to improved presentation.
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Alarcón, A., Forstnerič, F. & López, F.J. Embedded minimal surfaces in \({\mathbb {R}}^n\) . Math. Z. 283, 1–24 (2016). https://doi.org/10.1007/s00209-015-1586-5
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DOI: https://doi.org/10.1007/s00209-015-1586-5