Abstract
We show that for every conformal minimal immersion
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
[1]
A. Alarcón and I. Fernández,
Complete minimal surfaces in
[2] A. Alarcón, I. Fernández and F. J. López, Complete minimal surfaces and harmonic functions, Comment. Math. Helv. 87 (2012), 891–904. 10.4171/CMH/272Search in Google Scholar
[3]
A. Alarcón, I. Fernández and F. J. López,
Harmonic mappings and conformal minimal immersions of Riemann surfaces into
[4] A. Alarcón and F. Forstnerič, Null curves and directed immersions of open Riemann surfaces, Invent. Math. 196 (2014), 733–771. 10.1007/s00222-013-0478-8Search in Google Scholar
[5]
A. Alarcón, F. Forstnerič and F. J. López,
Embedded conformal minimal surfaces in
[6]
A. Alarcón and F. J. López,
Minimal surfaces in
[7] B. Drinovec Drnovšek and F. Forstnerič, Holomorphic curves in complex spaces, Duke Math. J. 139 (2007), 203–254. 10.1215/S0012-7094-07-13921-8Search in Google Scholar
[8] Y. Eliashberg and N. Mishachev, Introduction to the h-principle, Grad. Stud. Math. 48, American Mathematical Society, Providence 2002. 10.1090/gsm/048Search in Google Scholar
[9] F. Forstnerič, The Oka principle for multivalued sections of ramified mappings, Forum Math. 15 (2003), 309–328. 10.1515/form.2003.018Search in Google Scholar
[10] F. Forstnerič, Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis, Ergeb. Math. Grenzgeb. (3) 56, Springer, Berlin 2011. 10.1007/978-3-642-22250-4Search in 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. 10.5802/afst.1388Search in Google Scholar
[12] M. Gromov, Partial differential relations, Ergeb. Math. Grenzgeb. (3) 9, Springer, Berlin 1986. 10.1007/978-3-662-02267-2Search in Google Scholar
[13] M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), 851–897. 10.1090/S0894-0347-1989-1001851-9Search in Google Scholar
[14] R. C. Gunning and R. Narasimhan, Immersion of open Riemann surfaces, Math. Ann. 174 (1967), 103–108. 10.1007/BF01360812Search in Google Scholar
[15] M. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. 10.1090/S0002-9947-1959-0119214-4Search in Google Scholar
[16] L. Hörmander, An Introduction to complex analysis in several variables, 3rd ed., North-Holland Math. Library 7, North Holland, Amsterdam 1990. Search in Google Scholar
[17]
L. P. Jorge and F. Xavier,
A complete minimal surface in
[18] R. Kusner and N. Schmitt, The spinor representation of surfaces in space, preprint (1996), http://arxiv.org/abs/dg-ga/9610005. Search in Google Scholar
[19] F. J. López and A. Ros, On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991), 293–300. 10.4310/jdg/1214446040Search in Google Scholar
[20] R. Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, New York 1986. Search in Google Scholar
[21] S. Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. (2) 69 (1959), 327–344. 10.2307/1970186Search in Google Scholar
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