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The First Thirty Years of Andersén-Lempert Theory

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Abstract

In this paper we expose the impact of the fundamental discovery, made by Erik Andersen and László Lempert in 1992, that the group generated by shears is dense in the group of holomorphic automorphisms of a complex Euclidean space of dimension n > 1. In three decades since its publication, their groundbreaking work led to the discovery of several new phenomena and to major new results in complex analysis and geometry involving Stein manifolds and affine algebraic manifolds with many automorphisms. The aim of this survey is to present the focal points of these developments, with a view towards the future.

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References

  1. A. Abbondandolo, L. Arosio, J. E. Fornæss, P. Majer, H. Peters, J. Raissy and L. Vivas, A survey on non-autonomous basins in several complex variables, arXiv: 1311.3835 (2014).

  2. S. S. Abhyankar and T. T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math., 276 (1975), 148–166.

    MathSciNet  MATH  Google Scholar 

  3. F. Acquistapace, F. Broglia and A. Tognoli, A relative embedding theorem for Stein spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 507–522.

    MathSciNet  MATH  Google Scholar 

  4. A. Alarcón, Complete complex hypersurfaces in the ball come in foliations, J. Differential Geom., in press, arXiv: 1802.02004.

  5. A. Alarcón and F. Forstnerič, Every bordered Riemann surface is a complete proper curve in a ball, Math. Ann., 357 (2013), 1049–1070.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Alarcón and F. Forstnerič, Complete densely embedded complex lines in ℂ2, Proc. Amer. Math. Soc., 146 (2018), 1059–1067.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Alarcon and F. Forstnerič, A foliation of the ball by complete holomorphic discs, Math. Z., 296 (2020), 169–174.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Alarcón, F. Forstnerič and F. J. López, Minimal Surfaces from a Complex Analytic Viewpoint, Springer Monographs in Mathematics, Springer (Cham, 2021).

    Book  MATH  Google Scholar 

  9. A. Alarcón and J. Globevnik, Complete embedded complex curves in the ball of ℂ2 can have any topology, Anal. PDE, 10 (2017), 1987–1999.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Alarcóon, J. Globevnikand F. J. Lóopez, A construction of complete complex hypersurfaces in the ball with control on the topology, J. Reine Angew. Math., 751 (2019), 289–308.

    Article  MathSciNet  Google Scholar 

  11. A. Alarcoón and F. J. Lóopez, Complete bounded embedded complex curves in ℂ2, J. Eur. Math.Soc.(JEMS), 18 (2016), 1675–1705.

    Article  MathSciNet  Google Scholar 

  12. E. Andersén, Volume-preserving automorphisms of ℂn, Complex Variables Theory Appl., 14 (1990), 223–235.

    Article  MathSciNet  MATH  Google Scholar 

  13. E. Andersén, Complete vector fields on (ℂ*)n, Proc. Amer. Math. Soc., 128 (2000), 1079–1085.

    Article  MathSciNet  MATH  Google Scholar 

  14. E. Andersen and L. Lempert, On the group of holomorphic automorphisms of ℂn, Invent. Math., 110 (1992), 371–388.

  15. R. B. Andrist, Stein spaces characterized by their endomorphisms, Trans. Amer. Math. Soc., 363 (2011), 2341–2355.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. B. Andrist, The density property for Gizatullin surfaces with reduced degenerate fibre, J. Geom. Anal., 28 (2018), 2522–2538.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. B. Andrist, The density property for Calogero-Moser spaces, Proc. Amer. Math. Soc., 149 (2021), 4207–4218.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. B. Andrist, F. Forstneric, T. Ritter and E. F. Wold, Proper holomorphic embeddings into Stein manifolds with the density property, J. Anal. Math., 130 (2016), 135–150.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. B. Andrist and H. Kraft, Varieties characterized by their endomorphisms, Math. Res. Lett., 21 (2014), 225–233.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. B. Andrist and F. Kutzschebauch, The fibred density property and the automorphism group of the spectral ball, Math. Ann., 370 (2018), 917–936.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. B. Andrist, F. Kutzschebauch and P.-M. Poloni, The density property for Gizatullin surfaces completed by four rational curves, Proc. Amer. Math. Soc., 145 (2017), 5097–5108.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. B. Andrist and R. Ugolini, A new notion of tameness, J. Math. Anal. Appl., 472 (2019), 196–215.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. B. Andrist and E. F. Wold, Riemann surfaces in Stein manifolds with the density property, Ann. Inst. Fourier (Grenoble), 64 (2014), 681–697.

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Arosio and F. Lárusson, Chaotic holomorphic automorphisms of Stein manifolds with the volume density property, J. Geom. Anal., 29 (2019), 1744–1762.

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Arosio and F. Lárusson, Generic aspects of holomorphic dynamics on highly flexible complex manifolds, Ann. Mat. Pura Appl. (4), 199 (2020), 1697–1711.

    Article  MathSciNet  MATH  Google Scholar 

  26. I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch and M. Zaidenberg, Infinite transitivity on affine varieties, in: Birational Geometry, Rational Curves, and Arithmetic, Springer (New York, 2013), pp. 1–13.

    MATH  Google Scholar 

  27. I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch and M. Zaidenberg, Flexible varieties and automorphism groups, Duke Math. J., 162 (2013), 767–823.

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Baader, F. Kutzschebauch and E. F. Wold, Knotted holomorphic discs in ℂ2, J. Reine Angew. Math., 648 (2010), 69–73.

    MathSciNet  MATH  Google Scholar 

  29. E. Bedford and J. Smillie, Fatou-Bieberbach domains arising from polynomial automorphisms, Indiana Univ. Math. J., 40 (1991), 789–792.

    Article  MathSciNet  MATH  Google Scholar 

  30. E. Bedford and J. Smillie, Polynomial diffeomorphisms of ℂ2. II. Stable manifolds and recurrence, J. Amer. Math. Soc., 4 (1991), 657–679.

    MathSciNet  MATH  Google Scholar 

  31. H. Behnke and K. Stein, Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität, Math. Ann., 116 (1939), 204–216.

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Behnke and P. Thullen, Zur Theorie der Funktionen Mehrerer Komplexer Veränderlichen, Math. Ann., 109 (1934), 313–323.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. R. Bell and R. Narasimhan, Proper holomorphic mappings of complex spaces, in: Several Complex Variables, VI, Encyclopaedia Math. Sci., vol. 69, Springer (Berlin, 1990), pp. 1–38.

    Google Scholar 

  34. L. Bieberbach, Beispiel zweier ganzer Funktionen zweier komplexer Variablen, welche eine schlichte volumtreue Abbildung des ℝ4 auf einen Teil seiner selbst vermitteln, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl., 1933 (1933), 476–479.

    MATH  Google Scholar 

  35. E. Bishop, Mappings of partially analytic spaces, Amer. J. Math., 83 (1961), 209–242.

    Article  MathSciNet  MATH  Google Scholar 

  36. L. Boc Thaler, Fatou Components, Ph.D. dissertation, University of Ljubljana (2016), http://www.matknjiz.si/doktorati/2016/Boc-Thaler-14521-20.pdf.

  37. L. Boc Thaler and F. Forstnerič, A long ℂ2 without holomorphic functions, Anal. PDE, 9 (2016), 2031–2050.

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Bochner and W. T. Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press (Princeton, N.J., 1948).

    MATH  Google Scholar 

  39. S. Borell and F. Kutzschebauch, Non-equivalent embeddings into complex Euclidean spaces, Internat. J. Math., 17 (2006), 1033–1046.

    Article  MathSciNet  MATH  Google Scholar 

  40. F. Bracci, J. Raissy and B. Stensønes, Automorphisms of ℂk with an invariant nonrecurrent attracting Fatou component biholomorphic to ℂ × (ℂ*)k−1, J. Eur. Math. Soc. (JEMS), 23 (2021), 639–666.

    Article  MathSciNet  MATH  Google Scholar 

  41. D. Brotbek, On the hyperbolicity of general hypersurfaces, Publ. Math. Inst. Hautes Études Sci., 126 (2017), 1–34.

    Article  MathSciNet  MATH  Google Scholar 

  42. G. T. Buzzard and J. E. Fornæss, Complete holomorphic vector fields and time-1 maps, Indiana Univ. Math. J., 44 (1995), 1175–1182.

    MathSciNet  MATH  Google Scholar 

  43. G. T. Buzzard and J. E. Fornæss, An embedding of ℂ in ℂ2 with hyperbolic complement, Math. Ann., 306 (1996), 539–546.

    Article  MathSciNet  MATH  Google Scholar 

  44. G. T. Buzzard and F. Forstnerič, A Carleman type theorem for proper holomorphic embeddings, Ark. Mat., 35 (1997), 157–169.

    Article  MathSciNet  MATH  Google Scholar 

  45. G. T. Buzzard and F. Forstnerič, An interpolation theorem for holomorphic automorphisms of ℂn, J. Geom. Anal., 10 (2000), 101–108.

    Article  MathSciNet  MATH  Google Scholar 

  46. G. T. Buzzard and J. H. Hubbard, A Fatou-Bieberbach domain avoiding a neighborhood of a variety of codimension 2, Math. Ann., 316 (2000), 699–702.

    Article  MathSciNet  MATH  Google Scholar 

  47. S. Cantat, A. Regeta and J. Xie, Families of commuting automorphisms, and a characterization of the affine space. arXiv:1912.01567 (2019).

  48. M. Cerne and F. Forstnerič, Embedding some bordered Riemann surfaces in the affine plane, Math. Res. Lett., 9 (2002), 683–696.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Cerne and J. Globevnik, On holomorphic embedding of planar domains into ℂ2, J. Anal. Math., 81 (2000), 269–282.

    Article  MathSciNet  MATH  Google Scholar 

  50. S. Charpentier and L. Kosiński, Construction of labyrinths in pseudoconvex domains, Math. Z., 296 (2020), 1021–1025.

    Article  MathSciNet  MATH  Google Scholar 

  51. S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math., 133 (1974), 219–271.

    Article  MathSciNet  MATH  Google Scholar 

  52. G. De Vito, New examples of Stein manifolds with volume density property, Complex Anal. Synerg., 6 (2020), Paper No. 9, 10 pp.

  53. H. Derksen and F. Kutzschebauch, Nonlinearizable holomorphic group actions, Math. Ann., 311 (1998), 41–53.

    Article  MathSciNet  MATH  Google Scholar 

  54. P. G. Dixon and J. Esterle, Michael’s problem and the Poincare-Fatou-Bieberbach phenomenon, Bull. Amer. Math. Soc. (N.S.), 15 (1986), 127–187.

    Article  MathSciNet  MATH  Google Scholar 

  55. F. Docquier and H. Grauert, Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann., 140 (1960), 94–123.

    Article  MathSciNet  MATH  Google Scholar 

  56. F. Donzelli, Algebraic density property of Danilov-Gizatullin surfaces, Math. Z., 272 (2012), 1187–1194.

    Article  MathSciNet  MATH  Google Scholar 

  57. B. Drinovec Drnovsek, Complete proper holomorphic embeddings of strictly pseudo-convex domains into balls, J. Math. Anal. Appl., 431 (2015), 705–713.

    Article  MathSciNet  MATH  Google Scholar 

  58. B. Drinovec Drnovsek and F. Forstnerič, Strongly pseudoconvex domains as subvarieties of complex manifolds, Amer. J. Math., 132 (2010), 331–360.

    Article  MathSciNet  MATH  Google Scholar 

  59. A. Dubouloz, L. Moser-Jauslin and P.-M. Poloni, Inequivalent embeddings of the Koras-Russell cubic 3-fold, Michigan Math. J., 59 (2010), 679–694.

    Article  MathSciNet  MATH  Google Scholar 

  60. A. Dubouloz, L. Moser-Jauslin and P.-M. Poloni, Automorphism groups of certain rational hypersurfaces in complex four-space, in: Automorphisms in Birational and Affine Geometry, Springer Proc. Math. Stat., vol. 79, Springer (Cham, 2014), pp. 301–312.

    MATH  Google Scholar 

  61. Y. Eliashberg and M. Gromov, Embeddings of Stein manifolds of dimension n into the affine space of dimension 3n/2 + 1, Ann. of Math. (2), 136 (1992), 123–135.

    Article  MathSciNet  MATH  Google Scholar 

  62. P. Fatou, Sur certaines fonctions uniformes de deux variables, C. R. Acad. Sci., Paris, 175 (1922), 1030–1033.

    MATH  Google Scholar 

  63. P. Fatou, Sur les fonctions móeromorphes de deux variables, C. R. Acad. Sci., Paris, 175 (1922), 862–865.

    MATH  Google Scholar 

  64. K.-H. Fieseler, On complex affine surfaces with ℂ+-action, Comment. Math. Helv., 69 (1994), 5–27.

    Article  MathSciNet  MATH  Google Scholar 

  65. J. E. Fornaess, An increasing sequence of Stein manifolds whose limit is not Stein, Math. Ann., 223 (1976), 275–277.

    Article  MathSciNet  MATH  Google Scholar 

  66. J. E. Fornæss, Short ℂk, in: Complex Analysis in Several Variables—Memorial Conference of Kiyoshi Oka’s Centennial Birthday, Adv. Stud. Pure Math., vol. 42, Math. Soc. Japan (Tokyo, 2004), pp. 95–108.

    Google Scholar 

  67. J. E. Fornass and N. Sibony, Increasing sequences of complex manifolds, Math. Ann., 255 (1981), 351–360.

    Article  MathSciNet  MATH  Google Scholar 

  68. J. E. Fornass and N. Sibony, Complex Hóenon mappings in C2 and Fatou-Bieberbach domains, Duke Math. J., 65 (1992), 345–380.

    MathSciNet  Google Scholar 

  69. J. E. Fornaess and N. Sibony, The closing lemma for holomorphic maps, Ergodic Theory Dynam. Systems, 17 (1997), 821–837.

    Article  MathSciNet  MATH  Google Scholar 

  70. J. E. Fornæss and B. Stensønes, Stable manifolds of holomorphic hyperbolic maps, Internat. J. Math., 15 (2004), 749–758.

    Article  MathSciNet  MATH  Google Scholar 

  71. J. E. Fornaess and E. L. Stout, Spreading polydiscs on complex manifolds, Amer. J. Math., 99 (1977), 933–960.

    Article  MathSciNet  MATH  Google Scholar 

  72. O. Forster, Plongements des varieátáes de Stein, Comment. Math. Helv., 45 (1970), 170–184.

    Article  MathSciNet  MATH  Google Scholar 

  73. F. Forstnerič, Approximation by automorphisms on smooth submanifolds of ℂn, Math. Ann., 300 (1994), 719–738.

    Article  MathSciNet  MATH  Google Scholar 

  74. F. Forstnerič, A theorem in complex symplectic geometry, J. Geom. Anal., 5 (1995), 379–393.

    Article  MathSciNet  MATH  Google Scholar 

  75. F. Forstnerič, Actions of (ℝ, +) and (ℂ, +) on complex manifolds, Math. Z., 223 (1996), 123–153.

    MathSciNet  Google Scholar 

  76. F. Forstnerič, Interpolation by holomorphic automorphisms and embeddings in ℂn, J. Geom. Anal., 9 (1999), 93–117.

    Article  MathSciNet  MATH  Google Scholar 

  77. F. Forstnerič, Noncritical holomorphic functions on Stein manifolds, Acta Math., 191 (2003), 143–189.

    Article  MathSciNet  MATH  Google Scholar 

  78. F. Forstnerič, Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 56, Springer (Heidelberg, 2011).

    Book  MATH  Google Scholar 

  79. F. Forstnerič, Holomorphic families of long ℂ2’s, Proc.Amer. Math.Soc., 140 (2012), 2383–2389.

    Article  MathSciNet  MATH  Google Scholar 

  80. F. Forstnerič, Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 56, 2nd ed., Springer (Cham, 2017).

    Book  MATH  Google Scholar 

  81. F. Forstnerič, Holomorphic embeddings and immersions of Stein manifolds: a survey, in: Geometric Complex Analysis, Springer Proc. Math. Stat., vol. 246, Springer (Singapore, 2018), pp. 145–169.

    Chapter  Google Scholar 

  82. F. Forstnerič, Proper holomorphic immersions into Stein manifolds with the density property, J. Anal. Math., 139 (2019), 585–596.

    Article  MathSciNet  MATH  Google Scholar 

  83. F. Forstnerič, J. Globevnik and J.-P. Rosay, Nonstraightenable complex lines in ℂ2, Ark. Mat., 34 (1996), 97–101.

    Article  MathSciNet  MATH  Google Scholar 

  84. F. Forstneričand F. Laárusson, Oka properties of groups of holomorphic and algebraic automorphisms of complex affine space, Math. Res. Lett., 21 (2014), 1047–1067.

    Article  MathSciNet  Google Scholar 

  85. F. Forstnerič and J. Prezelj, Oka’s principle for holomorphic fiber bundles with sprays, Math. Ann., 317 (2000), 117–154.

    Article  MathSciNet  MATH  Google Scholar 

  86. F. Forstnerič and T. Ritter, Oka properties of ball complements, Math. Z., 277 (2014), 325–338.

    Article  MathSciNet  MATH  Google Scholar 

  87. F. Forstnerič and J.-P. Rosay, Approximation of biholomorphic mappings by automorphisms of ℂn, Invent. Math., 112 (1993), 323–349; Erratum: Invent. Math., 118 (1994), 573–574.

    Article  MathSciNet  MATH  Google Scholar 

  88. F. Forstnerič and E. F. Wold, Bordered Riemann surfaces in ℂ2, J. Math. Pures Appl. (9), 91 (2009), 100–114.

    Article  MathSciNet  MATH  Google Scholar 

  89. F. Forstnerič and E. F. Wold, Embeddings of infinitely connected planar domains into ℂ2, Anal. PDE, 6 (2013), 499–514.

    Article  MathSciNet  MATH  Google Scholar 

  90. F. Forstnerič and E. F. Wold, Fatou-Bieberbach domains in ℂn ℝk, Ark. Mat., 53 (2015), 259–270.

    Article  MathSciNet  MATH  Google Scholar 

  91. F. Forstnerič and E. F. Wold, Holomorphic families of Fatou-Bieberbach domains and applications to Oka manifolds, Math. Res. Lett., 27 (2020), 1697–1706.

    Article  MathSciNet  MATH  Google Scholar 

  92. F. Forstnerič and E. F. Wold, Runge tubes in Stein manifolds with the density property, Proc. Amer. Math.Soc., 148 (2020), 569–575.

    Article  MathSciNet  MATH  Google Scholar 

  93. T. Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 106–110.

    Article  MathSciNet  MATH  Google Scholar 

  94. M. H. Gizatullin, Quasihomogeneous affine surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1047–1071.

    MathSciNet  MATH  Google Scholar 

  95. J. Globevnik, A bounded domain in ℂN which embeds holomorphically into ℂN+1, Ark. Mat., 35 (1997), 313–325.

    Article  MathSciNet  MATH  Google Scholar 

  96. J. Globevnik, On Fatou-Bieberbach domains, Math. Z., 229 (1998), 91–106.

    Article  MathSciNet  MATH  Google Scholar 

  97. J. Globevnik, A complete complex hypersurface in the ball of ℂN, Ann. of Math. (2), 182 (2015), 1067–1091.

    Article  MathSciNet  MATH  Google Scholar 

  98. J. Globevnik, Holomorphic functions unbounded on curves of finite length, Math. Ann., 364 (2016), 1343–1359.

    Article  MathSciNet  MATH  Google Scholar 

  99. J. Globevnik and B. Stensønes, Holomorphic embeddings of planar domains into ℂ2, Math. Ann., 303 (1995), 579–597.

    Article  MathSciNet  MATH  Google Scholar 

  100. H. Grauert and R. Remmert, Theory of Stein spaces, Classics in Mathematics, Springer-Verlag, (Berlin, 2004).

    Book  MATH  Google Scholar 

  101. M. Gromov, Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 9, Springer-Verlag (Berlin, 1986).

    Book  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  103. Z.-X. He and O. Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. (2), 137 (1993), 369–406.

    Article  MathSciNet  MATH  Google Scholar 

  104. A. V. Isaev and N. G. Kruzhilin, Effective actions of the unitary group on complex manifolds, Canad. J. Math., 54 (2002), 1254–1279.

    Article  MathSciNet  MATH  Google Scholar 

  105. H. Iss’sa, On the meromorphic function field of a Stein variety, Ann. of Math. (2), 83 (1966), 34–46.

    Article  MathSciNet  MATH  Google Scholar 

  106. P. W. Jones, A complete bounded complex submanifold of ℂ3, Proc. Amer. Math. Soc., 76 (1979), 305–306.

    MathSciNet  Google Scholar 

  107. S. Kaliman, Extensions of isomorphisms of subvarieties in flexible varieties, Transform. Groups, 25 (2020), 517–575.

    Article  MathSciNet  MATH  Google Scholar 

  108. S. Kaliman, M. Koras, L. Makar-Limanov and P. Russell, ℂ*-actions on ℂ3 are linearizable, Electron. Res. Announc. Amer. Math. Soc., 3 (1997), 63–71.

    Article  MathSciNet  MATH  Google Scholar 

  109. S. Kaliman and F. Kutzschebauch, Criteria for the density property of complex manifolds, Invent. Math., 172 (2008), 71–87.

    Article  MathSciNet  MATH  Google Scholar 

  110. S. Kaliman and F. Kutzschebauch, Density property for hypersurfaces \(UV = P(\overline X)\), Math. Z., 258 (2008), 115–131.

    Article  MathSciNet  MATH  Google Scholar 

  111. S. Kaliman and F. Kutzschebauch, Algebraic volume density property of affine algebraic manifolds, Invent. Math., 181 (2010), 605–647.

    Article  MathSciNet  MATH  Google Scholar 

  112. S. Kaliman and F. Kutzschebauch, On the present state of the Andersóen-Lempert theory, in: Affine Algebraic Geometry, CRM Proc.Lecture Notes, vol. 54, Amer. Math. Soc. (Providence, RI, 2011), pp. 85–122.

    Chapter  Google Scholar 

  113. S. Kaliman and F. Kutzschebauch, On the density and the volume density property, in: Complex Analysis and Geometry, Springer Proc. Math. Stat., vol. 144, Springer (Tokyo, 2015), 175–186.

    Chapter  MATH  Google Scholar 

  114. S. Kaliman and F. Kutzschebauch, On algebraic volume density property, Transform. Groups, 21 (2016), 451–478.

    Article  MathSciNet  MATH  Google Scholar 

  115. S. Kaliman and F. Kutzschebauch, Algebraic (volume) density property for affine homogeneous spaces, Math. Ann., 367 (2017), 1311–1332.

    Article  MathSciNet  MATH  Google Scholar 

  116. S. Kaliman and L. Makar-Limanov, On the Russell-Koras contractible threefolds, J. Algebraic Geom., 6 (1997), 247–268.

    MathSciNet  MATH  Google Scholar 

  117. S. Kaliman and M. Zaidenberg, Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups, 4 (1999), 53–95.

    Article  MathSciNet  MATH  Google Scholar 

  118. K. Kodaira, Holomorphic mappings of polydiscs into compact complex manifolds, J. Differential Geometry, 6 (1971/72), 33–46.

    Article  MathSciNet  MATH  Google Scholar 

  119. H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn (Braunschweig, 1984).

    Book  MATH  Google Scholar 

  120. H. Kraft, Automorphism groups of affine varieties and a characterization of affine n-space, Trans. Moscow Math. Soc., 78 (2017), 171–186.

    Article  MathSciNet  MATH  Google Scholar 

  121. Y. Kusakabe, Oka properties of complements of holomorphically convex sets, arXiv: 2005.08247 (2020).

  122. Y. Kusakabe, Elliptic characterization and localization of Oka manifolds, Indiana Univ. Math. J., 70 (2021), 1039–1054.

    Article  MathSciNet  MATH  Google Scholar 

  123. F. Kutzschebauch, Andersáen-Lempert-theory with parameters: a representation theoretic point of view, J. Algebra Appl., 4 (2005), 325–340.

    Article  MathSciNet  MATH  Google Scholar 

  124. F. Kutzschebauch, Manifolds with infinite dimensional group of holomorphic automorphisms and the linearization problem, in: Handbook of Group Actions. V, Adv. Lect. Math. (ALM), vol. 48, Int. Press (Somerville, MA, 2020), pp. 257–300.

    MATH  Google Scholar 

  125. F. Kutzschebauch, F. Lárusson and G. W. Schwarz, Sufficient conditions for holomorphic linearisation, Transform. Groups, 22 (2017), 475–485.

    Article  MathSciNet  MATH  Google Scholar 

  126. F. Kutzschebauch, M. Leuenberger and A. Liendo, The algebraic density property for affine toric varieties, J. Pure Appl. Algebra, 219 (2015), 3685–3700.

    Article  MathSciNet  MATH  Google Scholar 

  127. F. Kutzschebauch, E. Lø w and E. F. S. Wold, Embedding some Riemann surfaces into ℂ2 with interpolation, Math. Z., 262 (2009), 603–611.

    Article  MathSciNet  Google Scholar 

  128. F. Kutzschebauch and P.-M. Poloni, Embedding Riemann surfaces with isolated punctures into the complex plane, Proc.Amer.Math.Soc., 148 (2020), 4831–4835.

    Article  MathSciNet  MATH  Google Scholar 

  129. F. Kutzschebauch and A. Ramos-Peon, An Oka principle for a parametric infinite transitivity property, J. Geom. Anal., 27 (2017), 2018–2043.

    Article  MathSciNet  MATH  Google Scholar 

  130. F. Kutzschebauch and G. W. Schwarz, A characterization of linearizability for holomorphic ℂ*-actions, Internat. Math. Res. Notices (2021).

  131. F. Kutzschebauch and E. F. Wold, Carleman approximation by holomorphic automorphisms of ℂn, J. Reine Angew. Math., 738 (2018), 131–148.

    Article  MathSciNet  MATH  Google Scholar 

  132. L. Leau, Étude sur les équations fonctionelles à une ou plusieurs variables, Toulouse Ann., 11 (1897), e25–e110.

    Article  MathSciNet  MATH  Google Scholar 

  133. M. Leuenberger, (Volume) density property of a family of complex manifolds including the Koras-Russell cubic threefold, Proc. Amer. Math. Soc., 144 (2016), 3887–3902.

    Article  MathSciNet  MATH  Google Scholar 

  134. E. Løw, J. V. Pereira, H. Peters and E. F. Wold, Polynomial completion of symplectic jets and surfaces containing involutive lines, Math. Ann., 364 (2016), 519–538.

    Article  MathSciNet  MATH  Google Scholar 

  135. I. Majcen, Embedding certain infinitely connected subsets of bordered Riemann surfaces properly into ℂ2, J. Geom. Anal., 19 (2009), 695–707.

    Article  MathSciNet  MATH  Google Scholar 

  136. I. Majcen, Proper holomorphic embeddings of finitely connected planar domains into ℂn, Ark. Mat., 51 (2013), 329–343.

    Article  MathSciNet  MATH  Google Scholar 

  137. L. Makar-Limanov, On the hypersurface x+x2y+z2+t3 = 0 in ℂ4 or a ℂ3-like threefold which is not ℂ3, Israel J. Math., 96 (1996), 419–429.

    Article  MathSciNet  Google Scholar 

  138. M. Miyanishi and T. Sugie, Affine surfaces containing cylinderlike open sets, J. Math. Kyoto Univ., 20 (1980), 11–42.

    MathSciNet  MATH  Google Scholar 

  139. L. Moser-Jauslin, Automorphism groups of Koras-Russell threefolds of the first kind, in: Affine Algebraic Geometry, CRM Proc. Lecture Notes, vol. 54, Amer. Math. Soc. (Providence, RI, 2011). pp. 261–270.

    Chapter  Google Scholar 

  140. R. Narasimhan, Imbedding of holomorphically complete complex spaces, Amer. J. Math., 82 (1960), 917–934.

    Article  MathSciNet  MATH  Google Scholar 

  141. Y. Nishimura, Automorphismes analytiques admettant des sous-variáetáes de points fixés attractives dans la direction transversale, J. Math. Kyoto Univ., 23 (1983), 289–299.

    MathSciNet  MATH  Google Scholar 

  142. Y. Nishimura, Applications holomorphes injectives de ℂ2 dans lui-même qui exceptent une droite complexe, J. Math. Kyoto Univ., 24 (1984), 755–761.

    MathSciNet  MATH  Google Scholar 

  143. K. Oka, Sur les fonctions analytiques de plusieurs variables. III. Deuxieme probleme de Cousin, J. Sci. Hiroshima Univ., Ser. A, 9 (1939), 7–19.

    MATH  Google Scholar 

  144. H. Peters, L. R. Vivas and E. F. Wold, Attracting basins of volume preserving automorphisms of ℂk, Internat. J. Math., 19 (2008), 801–810.

    Article  MathSciNet  MATH  Google Scholar 

  145. E. Picard, Sur une classe de surfaces algóebriques dont les coordonnóees s’expriment par des fonctions uniformes de deux paramètres, Bull. Soc. Math. Fr., 28 (1900), 17–25.

    Article  MATH  Google Scholar 

  146. E. Picard, Sur certaines eóquations fonctionnelles et sur une classe de surfaces algebriques, C. R. Acad. Sci., Paris, 139 (1905), 5–9.

    MATH  Google Scholar 

  147. H. Poincaróe, Sur une classe nouvelle de transcendantes uniformes, J. Math. Pures. Appl. (4), 6 (1890), 313–365.

    Google Scholar 

  148. H. Poincaróe, Sur l’uniformisation des fonctions analytiques, Acta Math., 31 (1907), 1–64.

    Article  MathSciNet  Google Scholar 

  149. C. P. Ramanujam, A topological characterisation of the affine plane as an algebraic variety, Ann. of Math. (2), 94 (1971), 69–88.

    Article  MathSciNet  MATH  Google Scholar 

  150. A. Ramos-Peon, Non-algebraic examples of manifolds with the volume density property, Proc.Amer. Math.Soc., 145 (2017), 3899–3914.

    Article  MathSciNet  MATH  Google Scholar 

  151. A. Ramos-Peon and R. Ugolini, Parametric jet interpolation for Stein manifolds with the density property, Internat. J. Math., 30 (2019), 1950046, 16 pp.

    Article  MathSciNet  MATH  Google Scholar 

  152. L. Reich, Das Typenproblem bei formal-biholomorphen Abbildungen mit anziehendem Fixpunkt, Math. Ann., 179 (1969), 227–250.

    Article  MathSciNet  MATH  Google Scholar 

  153. L. Reich, Normalformen biholomorpher Abbildungen mit anziehendem Fixpunkt, Math. Ann., 180 (1969), 233–255.

    Article  MathSciNet  MATH  Google Scholar 

  154. R. Remmert, Sur les espaces analytiques holomorphiquement sóeparables et holomorphiquement convexes, C. R. Acad. Sci. Paris, 243 (1956), 118–121.

    MathSciNet  MATH  Google Scholar 

  155. J.-P. Rosay and W. Rudin, Holomorphic maps from ℂn to ℂn, Trans. Amer. Math. Soc., 310 (1988), 47–86.

    MathSciNet  MATH  Google Scholar 

  156. J.-P. Rosay and W. Rudin, Holomorphic embeddings of ℂ in ℂn, in: Several Complex Variables (Stockholm, 1987/1988), Math. Notes, vol. 38, Princeton Univ. Press (Princeton, NJ, 1993), 563–569.

    Google Scholar 

  157. W. Rudin, Injective polynomial maps are automorphisms, Amer. Math. Monthly, 102 (1995), 540–543.

    Article  MathSciNet  MATH  Google Scholar 

  158. R. Schoen and S. T. Yau, Lectures on Harmonic Maps, Conference Proceedings and Lecture Notes in Geometry and Topology, II, International Press (Cambridge, MA, 1997).

    MATH  Google Scholar 

  159. J. Schurmann, Embeddings of Stein spaces into affine spaces of minimal dimension, Math. Ann., 307 (1997), 381–399.

    Article  MathSciNet  MATH  Google Scholar 

  160. I. R. Shafarevich, On some infinite-dimensional groups, Rend. Mat. e Appl. (5), 25 (1966), 208–212.

    MathSciNet  Google Scholar 

  161. I. R. Shafarevich, On some infinite-dimensional groups. II, Izv. Akad. Nauk SSSR Ser. Mat., 45 (1981), 214–226, 240 (in Russian).

    MathSciNet  MATH  Google Scholar 

  162. J.-L. Stehle, Plongements du disque dans ℂ2, in: Séminaire Pierre Lelong (Analyse), Annóee 1970–1971, Lecture Notes in Math., vol. 275, Springer (Berlin, 1972), pp. 119–130.

    Google Scholar 

  163. K. Stein, Überlagerungen holomorph-vollstäandiger komplexer Räume, Arch. Math. (Basel), 7 (1956), 354–361.

    Article  MathSciNet  MATH  Google Scholar 

  164. B. Stensønes, Fatou-Bieberbach domains with \({{\cal C}^\infty}\)-smooth boundary, Ann. of Math. (2), 145 (1997), 365–377.

    Article  MathSciNet  MATH  Google Scholar 

  165. S. Sternberg, Local contractions and a theorem of Poincaróe, Amer. J. Math., 79 (1957), 809–824.

    Article  MathSciNet  MATH  Google Scholar 

  166. G. Stolzenberg, The analytic part of the Runge hull, Math. Ann., 164 (1966), 286–290.

    Article  MathSciNet  MATH  Google Scholar 

  167. G. Stolzenberg, Uniform approximation on smooth curves, Acta Math., 115 (1966), 185–198.

    Article  MathSciNet  MATH  Google Scholar 

  168. E. L. Stout, Bounded holomorphic functions on finite Riemann surfaces, Trans. Amer. Math. Soc., 120 (1965), 255–285.

    MathSciNet  MATH  Google Scholar 

  169. M. Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algáebriques de l’espace ℂ2, J. Math. Soc. Japan, 26 (1974), 241–257.

    Article  MathSciNet  MATH  Google Scholar 

  170. Á. Tóth and D. Varolin, Holomorphic diffeomorphisms of complex semisimple Lie groups, Invent. Math., 139 (2000), 351–369.

    Article  MathSciNet  MATH  Google Scholar 

  171. R. Ugolini, A parametric jet-interpolation theorem for holomorphic automorphisms of ón, J. Geom. Anal., 27 (2017), 2684–2699.

    Article  MathSciNet  MATH  Google Scholar 

  172. D. Varolin, A general notion of shears, and applications, Michigan Math. J., 46 (1999), 533–553.

    Article  MathSciNet  MATH  Google Scholar 

  173. D. Varolin, The density property for complex manifolds and geometric structures. II, Internat. J. Math., 11 (2000), 837–847.

    Article  MathSciNet  MATH  Google Scholar 

  174. D. Varolin, The density property for complex manifolds and geometric structures, J. Geom. Anal., 11 (2001), 135–160.

    Article  MathSciNet  MATH  Google Scholar 

  175. J. Wermer, An example concerning polynomial convexity, Math. Ann., 139 (1959), 147–150 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  176. J. Winkelmann, Tame discrete subsets in Stein manifolds, J. Aust. Math. Soc., 107 (2019), 110–132.

    Article  MathSciNet  MATH  Google Scholar 

  177. E. F. Wold, Fatou-Bieberbach domains, Internat. J. Math., 16 (2005), 1119–1130.

    Article  MathSciNet  MATH  Google Scholar 

  178. E. F. Wold, Embedding Riemann surfaces properly into ℂ2, Internat. J. Math., 17 (2006), 963–974.

    Article  MathSciNet  MATH  Google Scholar 

  179. E. F. Wold, Proper holomorphic embeddings of finitely and some infinitely connected subsets of ℂ into ℂ2, Math. Z., 252 (2006), 1–9.

    Article  MathSciNet  Google Scholar 

  180. E. F. Wold, Embedding subsets of tori properly into ℂ2, Ann. Inst. Fourier (Grenoble), 57 (2007), 1537–1555.

    Article  MathSciNet  MATH  Google Scholar 

  181. E. F. Wold, A Fatou-Bieberbach domain in ℂ2 which is not Runge, Math. Ann., 340 (2008), 775–780.

    Article  MathSciNet  MATH  Google Scholar 

  182. E. F. Wold, A long ℂ2 which is not Stein, Ark. Mat., 48 (2010), 207–210.

    Article  MathSciNet  MATH  Google Scholar 

  183. E. F. Wold, A Fatou-Bieberbach domain intersecting the plane in the unit disk, Proc. Amer. Math. Soc., 140 (2012), 4205–4208.

    Article  MathSciNet  MATH  Google Scholar 

  184. P. Yang, Curvature of complex submanifolds of ℂn, in: Several Complex Variables, Proc. Sympos. Pure Math., vol. XXX, Part 2, Williams Coll. (Williamstown, Mass, 1975), pp. 135–137; Amer. Math. Soc. (Providence, RI, 1977).

    Google Scholar 

  185. P. Yang, Curvatures of complex submanifolds of ℂn, J. Differential Geom., 12 (1977), 499–511 (1978).

    Article  Google Scholar 

  186. M. Zaĭdenberg, Exotic algebraic structures on affine spaces, Algebra i Analiz, 11 (1999), 3–73 (in Russian); translation in St. Petersburg Math. J., 11 (2000), 703–760.

    MathSciNet  Google Scholar 

  187. R. Zentner, Integer homology 3-spheres admit irreducible representations in SL(2, ℂ), Duke Math. J., 167 (2018), 1643–1712.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors wish to thank Rafael Andrist, Shulim Kaliman, Finnur Larusson, and Riccardo Ugolini for their helpful remarks and suggestions.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia

    F. Forstnerič

  2. Institute of Mathematics, University of Bern, Sidlerstrasse 5, CH-3012, Bern, Switzerland

    F. Kutzschebauch

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  1. F. Forstnerič
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Correspondence to F. Forstnerič.

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Dedicated to László Lempert in honour of his 70th birthday

Forstneric is supported by research program P1-0291 and grant J1-3005 from ARRS, Republic of Slovenia.

Kutzschebauch is supported by Schweizerische Nationalfonds Grant Nr. 200021-178730.

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Forstnerič, F., Kutzschebauch, F. The First Thirty Years of Andersén-Lempert Theory. Anal Math 48, 489–544 (2022). https://doi.org/10.1007/s10476-022-0130-1

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