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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

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1435-5345
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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

## Abstract

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

## References

• [1]

A. Alarcón and I. Fernández, Complete minimal surfaces in ${ℝ}^{3}$ with a prescribed coordinate function, Differential Geom. Appl. 29 (2011), no. 1, S9–S15. Google Scholar

• [2]

A. Alarcón, I. Fernández and F. J. López, Complete minimal surfaces and harmonic functions, Comment. Math. Helv. 87 (2012), 891–904.

• [3]

A. Alarcón, I. Fernández and F. J. López, Harmonic mappings and conformal minimal immersions of Riemann surfaces into ${ℝ}^{N}$, Calc. Var. Partial Differential Equations 47 (2013), 227–242. Google Scholar

• [4]

A. Alarcón and F. Forstnerič, Null curves and directed immersions of open Riemann surfaces, Invent. Math. 196 (2014), 733–771.

• [5]

A. Alarcón, F. Forstnerič and F. J. López, Embedded conformal minimal surfaces in ${ℝ}^{n}$, preprint (2014), http://arxiv.org/abs/1409.6901; Math. Z. (2015), DOI 10.1007/s00209-015-1586-5.

• [6]

A. Alarcón and F. J. López, Minimal surfaces in ${ℝ}^{3}$ properly projecting into ${ℝ}^{2}$, J. Differential Geom. 90 (2012), 351–382. Google Scholar

• [7]

B. Drinovec Drnovšek and F. Forstnerič, Holomorphic curves in complex spaces, Duke Math. J. 139 (2007), 203–254.

• [8]

Y. Eliashberg and N. Mishachev, Introduction to the h-principle, Grad. Stud. Math. 48, American Mathematical Society, Providence 2002. Google Scholar

• [9]

F. Forstnerič, The Oka principle for multivalued sections of ramified mappings, Forum Math. 15 (2003), 309–328. Google Scholar

• [10]

F. Forstnerič, Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis, Ergeb. Math. Grenzgeb. (3) 56, Springer, Berlin 2011. Google Scholar

• [11]

F. Forstnerič, Oka manifolds: From Oka to Stein and back. With an appendix by F. Lárusson, Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 747–809.

• [12]

M. Gromov, Partial differential relations, Ergeb. Math. Grenzgeb. (3) 9, Springer, Berlin 1986. Google Scholar

• [13]

M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), 851–897. Google Scholar

• [14]

R. C. Gunning and R. Narasimhan, Immersion of open Riemann surfaces, Math. Ann. 174 (1967), 103–108.

• [15]

M. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276.

• [16]

L. Hörmander, An Introduction to complex analysis in several variables, 3rd ed., North-Holland Math. Library 7, North Holland, Amsterdam 1990. Google Scholar

• [17]

L. P. Jorge and F. Xavier, A complete minimal surface in ${ℝ}^{3}$ between two parallel planes, Ann. of Math. (2) 112 (1980), 203–206. Google Scholar

• [18]

R. Kusner and N. Schmitt, The spinor representation of surfaces in space, preprint (1996), http://arxiv.org/abs/dg-ga/9610005.

• [19]

F. J. López and A. Ros, On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991), 293–300.

• [20]

R. Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, New York 1986. Google Scholar

• [21]

S. Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. (2) 69 (1959), 327–344.

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,

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