Every Meromorphic Function is the Gauss Map of a Conformal Minimal Surface

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Abstract

Let M be an open Riemann surface. We prove that every meromorphic function on M is the complex Gauss map of a conformal minimal immersion \(M\rightarrow \mathbb {R}^3\) which may furthermore be chosen as the real part of a holomorphic null curve \(M\rightarrow \mathbb {C}^3.\) Analogous results are proved for conformal minimal immersions \(M\rightarrow \mathbb {R}^n\) for any \(n>3.\) We also show that every conformal minimal immersion \(M\rightarrow \mathbb {R}^n\) is isotopic through conformal minimal immersions \(M\rightarrow \mathbb {R}^n\) to a flat one, and we identify the path connected components of the space of all conformal minimal immersions \(M\rightarrow \mathbb {R}^n\) for any \(n\ge 3.\)

Keywords

Riemann surface Complex curve Minimal surface Gauss map 

Mathematics Subject Classification

49Q05 30F99 
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Notes

Acknowledgements

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 Grant No. MTM2014-52368-P, Spain. F. Forstneric̆ is partially supported by the Research Program P1-0291 and Grants J1-5432 and J1-7256 from ARRS, Republic of Slovenia. A part of the work on this paper was done while F. Forstnerič was visiting the Department of Geometry and Topology of the University of Granada, Spain, in March 2016. He wishes to thank this institution for the invitation and hospitality.

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Authors and Affiliations

  1. 1.Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR)Universidad de GranadaGranadaSpain
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

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