Volume 22, issue 1 (2018)

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Complete minimal surfaces densely lying in arbitrary domains of $\mathbb{R}^n$

Antonio Alarcón and Ildefonso Castro-Infantes

Geometry & Topology 22 (2018) 571–590

In this paper we prove that, given an open Riemann surface M and an integer n 3, the set of complete conformal minimal immersions M n with X(M)̄ = n forms a dense subset in the space of all conformal minimal immersions M n endowed with the compact-open topology. Moreover, we show that every domain in n contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface.

Our method of proof can be adapted to give analogous results for nonorientable minimal surfaces in n(n 3), complex curves in n(n 2), holomorphic null curves in n(n 3), and holomorphic Legendrian curves in 2n+1(n ).

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complete minimal surface, Riemann surface, holomorphic curve
Mathematical Subject Classification 2010
Primary: 49Q05
Secondary: 32H02
Received: 15 November 2016
Accepted: 28 April 2017
Published: 31 October 2017
Proposed: Tobias H Colding
Seconded: Dmitri Burago, Leonid Polterovich
Antonio Alarcón
Departamento de Geometría y Topología
Universidad de Granada
Ildefonso Castro-Infantes
Departamento de Geometría y Topología
Universidad de Granada