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Journal für die reine und angewandte Mathematik

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Volume 2018, Issue 740


Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve

Antonio Alarcón
  • Departamento de Geometría y Topología and Instituto de Matemáticas IEMath-GR, Universidad de Granada, E-18071 Granada, Spain
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/ Franc Forstnerič
  • Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
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Published Online: 2015-12-11 | DOI: https://doi.org/10.1515/crelle-2015-0069


We show that for every conformal minimal immersion u:M3 from an open Riemann surface M to 3 there exists a smooth isotopy ut:M3 (t[0,1]) of conformal minimal immersions, with u0=u, such that u1 is the real part of a holomorphic null curve M3 (i.e. u1 has vanishing flux). If furthermore u is nonflat, then u1 can be chosen to have any prescribed flux and to be complete.


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About the article

Received: 2014-08-22

Revised: 2015-06-28

Published Online: 2015-12-11

Published in Print: 2018-07-01

Antonio Alarcón is supported by the Ramón y Cajal program of the Spanish Ministry of Economy and Competitiveness, and is partially supported by the MINECO/FEDER grants MTM2011-22547 and MTM2014-52368-P, Spain. Franc Forstnerič is supported in part by the research program P1-0291 and the grant J1-5432 from ARRS, Republic of Slovenia.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 740, Pages 77–109, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2015-0069.

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A. Alarcón, F. Forstneric̆, and F. J. López
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