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A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (ℂ*)2

  • Tyson Ritter EMAIL logo

Abstract

Let X be an open Riemann surface. We prove an Oka property on the approximation and interpolation of continuous maps X(*)2 by proper holomorphic embeddings, provided that we permit a smooth deformation of the complex structure on X outside a certain set. This generalises and strengthens a recent result of Alarcón and López. We also give a Forstnerič–Wold theorem for proper holomorphic embeddings (with respect to the given complex structure) of certain open Riemann surfaces into (*)2.

Award Identifier / Grant number: NFR-209751/F20

Funding statement: The author was supported by Norwegian Research Council grant NFR-209751/F20.

Acknowledgements

I wish to thank Erlend Wold and Erik Løw for helpful discussions during the preparation of this paper. I also thank Finnur Lárusson for his constructive comments on an earlier draft. I am grateful to the anonymous referee for their detailed feedback and suggestions for improving the exposition.

References

[1] A. Alarcón and F. López, Proper holomorphic embeddings of Riemann surfaces with arbitrary topology into 2, J. Geom. Anal. 23 (2013), 1794–1805. 10.1007/s12220-012-9306-4Search in Google Scholar

[2] E. Andersén, Volume-preserving automorphisms of n, Complex Variables Theory Appl. 14 (1990), 223–235. 10.1080/17476939008814422Search in Google Scholar

[3] E. Andersén and L. Lempert, On the group of holomorphic automorphisms of n, Invent. Math. 110 (1992), 371–388. 10.1007/BF01231337Search in Google Scholar

[4] M. Černe and F. Forstnerič, Embedding some bordered Riemann surfaces in the affine plane, Math. Res. Lett. 9 (2002), 683–696. 10.4310/MRL.2002.v9.n5.a10Search in Google Scholar

[5] F. Forstnerič, Stein manifolds and holomorphic mappings, Ergeb. Math. Grenzgeb. (3), Springer, Berlin 2011. 10.1007/978-3-642-22250-4Search in Google Scholar

[6] F. Forstnerič, Oka manifolds: From Oka to Stein and back, Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), 747–809. 10.5802/afst.1388Search in Google Scholar

[7] F. Forstnerič and F. Lárusson, Survey of Oka theory, New York J. Math. 17A (2011), 11–38. Search in Google Scholar

[8] F. Forstnerič, E. Løw and N. Øvrelid, Solving the d- and ¯-equations in thin tubes and applications to mappings, Michigan Math. J. 49 (2001), 369–416. 10.1307/mmj/1008719779Search in Google Scholar

[9] F. Forstnerič and J.-P. Rosay, Approximation of biholomorphic mappings by automorphisms of n, Invent. Math. 112 (1993), 323–349. 10.1007/BF01232438Search in Google Scholar

[10] F. Forstnerič and J.-P. Rosay, Erratum: “Approximation of biholomorphic mappings by automorphisms of n”, Invent. Math. 118 (1994), 573–574. 10.1007/BF01231544Search in Google Scholar

[11] F. Forstnerič and M. Slapar, Deformations of Stein structures and extensions of holomorphic mappings, Math. Res. Lett. 14 (2007), 343–357. 10.4310/MRL.2007.v14.n2.a15Search in Google Scholar

[12] F. Forstnerič and M. Slapar, Stein structures and holomorphic mappings, Math. Z. 256 (2007), 615–646. 10.1007/s00209-006-0093-0Search in Google Scholar

[13] F. Forstnerič and E. F. Wold, Bordered Riemann surfaces in 2, J. Math. Pures Appl. (9) 91 (2009), 100–114. 10.1016/j.matpur.2008.09.010Search in Google Scholar

[14] J. Globevnik and B. Stensønes, Holomorphic embeddings of planar domains into 2, Math. Ann. 303 (1995), 579–597. 10.1007/BF01461006Search in Google Scholar

[15] 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

[16] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. 2, John Wiley & Sons, New York 1969. Search in Google Scholar

[17] F. Kutzschebauch, E. Løw and E. F. Wold, Embedding some Riemann surfaces into 2 with interpolation, Math. Z. 262 (2009), 603–611. 10.1007/s00209-008-0392-8Search in Google Scholar

[18] F. Lárusson and T. Ritter, Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into ×*, Indiana Univ. Math. J. 63 (2014), no. 2, 367–383. 10.1512/iumj.2014.63.5206Search in Google Scholar

[19] J. Milnor, Morse theory, Ann. of Math. Stud. 51, Princeton University Press, Princeton 1963. 10.1515/9781400881802Search in Google Scholar

[20] T. Ritter, A strong Oka principle for embeddings of some planar domains into ×*, J. Geom. Anal. 23 (2013), 571–597. 10.1007/s12220-011-9254-4Search in Google Scholar

[21] D. Varolin, The density property for complex manifolds and geometric structures. II, Internat. J. Math. 11 (2000), 837–847. 10.1142/S0129167X00000404Search in Google Scholar

[22] D. Varolin, The density property for complex manifolds and geometric structures, J. Geom. Anal. 11 (2001), 135–160. 10.1007/BF02921959Search in Google Scholar

[23] E. F. Wold, Embedding Riemann surfaces properly into 2, Internat. J. Math. 17 (2006), 963–974. 10.1142/S0129167X06003746Search in Google Scholar

[24] E. F. Wold, Proper holomorphic embeddings of finitely and some infinitely connected subsets of into 2, Math. Z. 252 (2006), 1–9. 10.1007/s00209-005-0836-3Search in Google Scholar

Received: 2014-05-09
Revised: 2015-06-01
Published Online: 2016-04-16
Published in Print: 2018-12-01

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