Embedded minimal surfaces in $${\mathbb {R}}^n$$

Mathematische Zeitschrift - Tập 283 - Trang 1-24 - 2015
Antonio Alarcón1, Franc Forstnerič2,3, Francisco J. López4
1Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Granada, Spain
2Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
3Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
4Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain

Tóm tắt

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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Mathematische Zeitschrift

Tập 283

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