Acta Mathematica

Noncritical holomorphic functions on Stein manifolds

Franc Forstnerič

Full-text: Open access

Note

In memory of my father, Franc, and sister, Helena

Article information

Source
Acta Math. Volume 191, Number 2 (2003), 143-189.

Dates
Received: 7 August 2002
First available in Project Euclid: 31 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.acta/1485891605

Digital Object Identifier
doi:10.1007/BF02392963

Zentralblatt MATH identifier
1064.32021

Rights
2003 © Institut Mittag-Leffler

Citation

Forstnerič, Franc. Noncritical holomorphic functions on Stein manifolds. Acta Math. 191 (2003), no. 2, 143--189. doi:10.1007/BF02392963. http://projecteuclid.org/euclid.acta/1485891605.


Export citation

References

  • [A] Andersén, E., Volume-preserving automorphisms of Cn. Complex Variables Theory Appl., 14 (1990), 223–235.
  • [AL] Andersén, E. & Lempert, L., On the group of holomorphic automorphisms of Cn. Invent. Math., 110 (1992), 371–388.
  • [AF] Andreotti, A. & Frankel, T., The Lefschetz theorem on hyperplane sections. Ann. of Math. (2), 69 (1959), 713–717.
  • [BN] Bell, S. R. & Narasimhan, R., Proper holomorphic mappings of complex spaces, in Several Complex Variables, Vol. VI, pp. 1–38, Encyclopaedia Math. Sci., 69, Springer-Verlag, Berlin, 1990.
  • [DG] Docquier, F & Graubert, H., Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann., 140 (1960), 94–123.
  • [EM] Eliashberg, Y. & Mishachev, N., Introduction to the h-Principle. Graduate Studies in Math., 48. Amer. Math. Soc., Providence, RI, 2002.
  • [Fe] Feit, S. D., k-mersions of manifolds. Acta Math., 122 (1969), 173–195.
  • [Fo1] Forster, O., Some remarks on parallelizable Stein manifolds. Bull. Amer. Math. Soc., 73 (1967), 712–716.
  • [Fo2] —, Plongements des variétés de Stein. Comment. Math. Helv., 45 (1970), 170–184.
  • [F1] Forstnerič, F., Approximation by automorphisms on smooth submanifolds of Cn. Math. Ann., 300 (1994), 719–738.
  • [F2] —, Interpolation by holomorphic automorphisms and embeddings in Cn. J. Geom. Anal., 9 (1999), 93–117.
  • [F3] —, The homotopy principle in complex analysis: A survey, in Explorations in Complex and Riemannian Geometry: A Volume Dedicated to Robert E. Greene (J. Bland, K.-T. Kim and S. G. Krantz, eds.), pp. 73–99. Contemporary Mathematics, 332. Amer. Math. Soc., Providence, RI, 2003.
  • [FL] Forstnerič, F. & Løw, E., Global holomorphic equivalence of smooth submanifolds in Cn. Indiana Univ. Math. J., 46 (1997), 133–153.
  • [FLØ] Forstnerič, F., Løw, E. & Øvrelid, N., Solving the d- and $\bar \partial $ -equations in thin tubes and applications to mappings. Michigan Math. J., 49 (2001), 369–416.
  • [FP1] Forstnerič, F. & Prezelj, J., Oka’s principle for holomorphic fiber bundles with sprays. Math. Ann., 317 (2000), 117–154.
  • [FP2]—, Extending holomorphic sections from complex subvarieties. Math. Z., 236 (2001), 43–68.
  • [FP3]—, Oka’s principle for holomorphic submersions with sprays. Math. Ann., 322 (2002), 633–666.
  • [FR] Forstnerič, F. & Rosay, J.-P., Approximation of biholomorphic mappings by automorphisms of Cn. Invent. Math., 112 (1993), 323–349; Erratum in Invent. Math., 118 (1994), 573–574.
  • [God] Godbillon, C., Feuilletages. Études géométriques. Progr. Math., 98. Birkhäuser, Basel, 1991.
  • [GG] Golubitsky, M. & Guillemin, V., Stable Mappings and their Singularities. Graduate Texts in Math., 14. Springer-Verlag, New York-Heidelberg, 1973.
  • [Gra] Grauert, H., Analytische Faserungen über holomorph-vollständigen Räumen. Math. Ann., 135 (1958), 263–273.
  • [Gro1] Gromov, M., Stable mappings of foliations into manifolds. Izv. Akad. Nauk SSSR Ser. Mat., 33 (1969), 707–734 (Russian).
  • [Gro2]—, Convex integration of differential relations, I. Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 329–343 (Russian); English translation in Math. USSR-Izv., 7 (1973), 329–343.
  • [Gro3]—, Partial Differential Relations, Ergeb. Math. Grenzgeb. (3), 9. Springer-Verlag, Berlin, 1986.
  • [Gro4]—, Oka’s principle for holomorphic sections of elliptic bundles. J. Amer. Math. Soc., 2 (1989), 851–897.
  • [GN] Gunning, R. C. & Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann., 174 (1967), 103–108.
  • [GR] Gunning, R. C. & Rossi, H., Analytic Functions of Several Complex Variables. Prentice-Hall, Englewood Cliffs, NJ, 1965.
  • [Ha1] Haefliger, A., Variétés feuilletés. Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 367–398
  • [Ha2]—, Lectures on the theorem of Gromov, in Proceedings of Liverpool Singularities Symposium, Vol. II (1969/1970), pp. 128–141. Lecture Notes in Math., 209, Springer-Verlag, Berlin, 1971.
  • [HW1] Harvey, F. R. & Wells, R. O., Jr., Holomorphic approximation and hyperfunction theory on a C1 totally real submanifold of a complex manifold. Math. Ann., 197 (1972), 287–318.
  • [HW2]—, Zero sets of non-negative strictly plurisubharmonic functions. Math. Ann., 201 (1973), 165–170.
  • [HL1] Henkin, G. M. & Leiterer, J., Theory of Functions on Complex Manifolds. Akademie-Verlag, Berlin, 1984.
  • [HL2]—, Andreotti-Grauert Theory by Integral Formulas. Progr. Math., 74 Birkhäuser Boston, Boston, 1988.
  • [HL3]—, The Oka-Grauert principle without induction over the base dimension. Math. Ann., 311 (1998), 71–93.
  • [Hi1] Hirsch, M. W., Immersions of manifolds. Trans. Amer. Math. Soc., 93 (1959), 242–276.
  • [Hi2]—, On imbedding differential manifolds in euclidean space. Ann. of Math. (2), 73 (1961), 566–571.
  • [Hö1] Hörmander, L., L2 estimates and existence theorems for the $\bar \partial $ operator. Acta Math., 113 (1965), 89–152.
  • [Hö2]—, An Introduction to Complex Analysis in Several Variables, 3rd edition. North-Holland Math. Library, 7. North-Holland, Amsterdam, 1990.
  • [HöW] Hörmander, L. & Wermer, J., Uniform approximations on compact sets in Cn. Math. Scand., 23 (1968), 5–21.
  • [Ko] Kolmogorov, A., On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables. Dokl. Akad. Nauk SSSR (N.S.), 108 (1956), 179–182 (Russian).
  • [Ku] Kutzschebauch, F., Andersén-Lempert theory with parameters. Preprint, 2002.
  • [LV] Lehto, O. & Virtanen, K. I., Quasiconformal Mappings in the Plane, 2nd edition. Grundlehren Math. Wiss., 126. Springer-Verlag, New York-Heidelberg, 1973.
  • [MS] Milnor, J. W. & Stasheff, J. D., Characteristic Classes. Ann. of Math. Stud., 76. Princeton Univ. Press, Princeton, NJ, 1974.
  • [N] Nishimura, Y., Examples of analytic immersions of two-dimensional Stein manifolds into C2. Math. Japon., 26 (1981), 81–83.
  • [Pf] Pfluger, A., Über die Konstruktion Riemannscher Flächen durch Verheftung. J. Indian Math. Soc. (N.S.), 24 (1960), 401–412.
  • [Ph1] Phillips, A., Submersions of open manifolds. Topology, 6 (1967), 171–206.
  • [Ph2]—, Foliations on open manifolds, I. Comment. Math. Helv., 43 (1968), 204–211.
  • [Ph3]—, Foliations on open manifolds, II. Comment. Math. Helv., 44 (1969), 367–370.
  • [Ph4]—, Smooth maps transverse to a foliation. Bull. Amer. Math. Soc., 76 (1970), 792–797.
  • [Ph5]—, Smooth maps of constant rank. Bull. Amer. Math. Soc., 80 (1974), 513–517.
  • [Ra] Ramspott, K. J., Stetige und holomorphe Schnitte in Bündeln mit homogener Faser. Math. Z., 89 (1965), 234–246.
  • [RS] Range, R. M. & Siu, Y. T., Ck approximation by holomorphic functions and $\bar \partial $ -closed forms on Ck submanifolds of a complex manifold. Math. Ann., 210 (1974), 105–122.
  • [Ro] Rosay, J. P., A counterexample related to Hartogs’ phenomenon (a question by E. Chirka). Michigan Math. J., 45 (1998), 529–535.
  • [Sm] Smale, S., The classification of immersions of spheres in Euclidean spaces. Ann. of Math. (2), 69 (1959), 327–344.
  • [Sp] Spring, D., Convex Integration Theory. Solutions to the h-Principle in Geometry and Topology. Monogr. Math., 92. Birkhäuser, Basel, 1998.
  • [Ste] Stein, K., Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem. Math. Ann., 123 (1951), 201–222.
  • [Sto] Stout, E. L., The Theory of Uniform Alglebras. Bogden & Quigley, Tarrytown-on-Hudson, NY, 1971.
  • [Tho] Thom, R., Un lemme sur les applications différentiables. Bol. Soc. Mat. Mexicana (2), 1 (1956), 59–71.
  • [Th1] Thurston, W., The theory of foliations of codimension greater than one. Comment. Math. Helv., 49 (1974), 214–231.
  • [Th2]—, Existence of codimension-one foliations. Ann. of Math. (2), 104 (1976), 249–268.
  • [V] Varolin, D., The density property for complex manifolds and geometric structures. J. Geom. Anal., 11 (2001), 135–160.