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Licensed Unlicensed Requires Authentication Published by De Gruyter December 11, 2015

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

  • Antonio Alarcón EMAIL logo and Franc Forstnerič

Abstract

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.

Funding statement: 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.

Acknowledgements

The authors wish to thank Jaka Smrekar for his help with the topological matters in Section 8. We also thank the referee for useful suggestions which lead to improved presentation.

References

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Received: 2014-08-22
Revised: 2015-06-28
Published Online: 2015-12-11
Published in Print: 2018-07-01

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