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Stationary holomorphic discs and finite jet determination problems

Mathematische Annalen volume 358, pages 477–509 (2014)Cite this article

Abstract

We construct a family of small analytic discs attached to Levi non-degenerate hypersurfaces in \(\mathbb{C }^{n+1}\), which is globally biholomorphically invariant. We then apply this technique to study unique determination problems along Levi non-degenerate hypersurfaces that are merely of class \(\mathcal{C }^4\). This method gives 2-jet determination results for germs of biholomorphisms, CR diffeomorphisms, as well as in the almost complex setting.

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References

  1. Baoudendi, M.S., Ebenfelt, P., Rothschild, L.P.: Parametrization of local biholomorphisms of real-analytic hypersurfaces. Asian J. Math. 1, 1–16 (1997)

    MathSciNet  Google Scholar 

  2. Baoudendi, M.S., Ebenfelt, P., Rothschild, L.P.: CR automorphisms of real analytic CR manifolds in complex space. Commun. Anal. Geom. 6, 291–315 (1998)

    Google Scholar 

  3. Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47. Princeton University Press, Princeton, NJ (1999)

  4. Baoudendi, M.S., Ebenfelt, P., Rothschild, L.P.: Convergence and finite determination of formal CR mappings. J. Am. Math. Soc. 13, 697–723 (2000)

    Article  Google Scholar 

  5. Baoudendi, M.S., Ebenfelt, P., Rothschild, L.P.: Local geometric properties of real submanifolds in complex space. Bull. Am. Math. Soc. 37, 309–336 (2000)

    Article  Google Scholar 

  6. Beloshapka, V.K.: A uniqueness theorem for automorphisms of a nondegenerate surface in a complex space. Mat. Zametki 47, 17–22 (1990) [Russian; translation in. Math. Notes 47, 239–242 (1990)]

    Google Scholar 

  7. Birkhoff, G.D.: A theorem on matrices of analytic functions. Math. Ann. 74, 122–133 (1913)

    Article  MathSciNet  Google Scholar 

  8. Blanc-Centi, L.: Regularity and estimates for J-holomorphic discs attached to a maximal totally real submanifold. J. Math. Anal. Appl. 341, 170–187 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Blanc-Centi, L.: Stationary discs glued to a Levi non-degenerate hypersurface. Trans. Am. Math. Soc. 361, 3223–3239 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Burns, D., Krantz, S.: Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J. Am. Math. Soc. 7, 661–676 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cartan, E.: Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, I. Ann. Math. Pura Appl. 11, 17–90 (1932) [Œuvres complètes, Part. II, Vol. 2, Gauthier-Villars, 1231–1304 (1952)]

  12. Cartan, E.: Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, II. Ann. Sci. Norm. Sup. Pisa 1, 333–354 (1932) [Œuvres complètes, Part III, vol. 2. Gauthier-Villars, pp. 1217–1238 (1952)]

  13. Cartan, H.: Sur les groupes de transformations analytiques. Act. Sci et Int, Hermann, Paris (1935)

  14. Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chirka, E.M.: Regularity of the boundaries of analytic sets. Mat. Sb. 45, 291–336 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  16. Clancey, K., Gohberg, I.: Factorization of matrix functions and singular integral operators. Birkhäuser Verlag, Basel (1981)

    Book  MATH  Google Scholar 

  17. Coupet, B., Gaussier, H., Sukhov, A.: Riemann maps in almost complex manifolds. Ann. Sci. Norm. Super. Pisa Cl. Sci. 2(5), 761–785 (2003)

    MATH  MathSciNet  Google Scholar 

  18. Coupet, B., Gaussier, H., Sukhov, A.: Fefferman’s mapping theorem on almost complex manifolds in complex dimension two. Math. Z. 250, 59–90 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ebenfelt, P.: Finite jet determination of holomorphic mappings at the boundary. Asian J. Math. 5, 637–662 (2001)

    MATH  MathSciNet  Google Scholar 

  20. Ebenfelt, P., Lamel, B., Zaitsev, D.: Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case. Geom. Funct. Anal. 13, 546–573 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gaussier, H., Sukhov, A.: On the geometry of model almost complex manifolds with boundary. Math. Z. 254, 567–589 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Globevnik, J.: Perturbation by analytic discs along maximal real submanifolds of \({\mathbb{C}}^N\). Math. Z. 217, 287–316 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  23. Globevnik, J.: Perturbing analytic discs attached to maximal totally real submanifolds of \({\mathbb{C}}^N\). Indag. Math. 7, 37–46 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  24. Han, C.K.: Analyticity of CR equivalences between real hypersurfaces in \({\mathbb{C}}^n\) with degenerate Levi form. Invent. Math. 73, 51–69 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  25. Han, C.K.: Complete system for the mappings of CR manifolds of nondegenerate Levi forms. Math. Ann. 309, 401–409 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hill, D.C., Taiani, G.: Families of analytic discs in \({\mathbb{C}}^n\) with boundaries in a prescribed CR manifold. Ann. Sci. Norm. Super. Pisa Cl. Sci. (4) 5, 327–380 (1978)

    Google Scholar 

  27. Huang, X.: A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains. Can. J. Math. 47, 405–420 (1995)

    Article  MATH  Google Scholar 

  28. Ivashkovich, S., Rosay, J.-P.: Schwarz-type lemmas for solutions of \(\bar{\partial }\)-inequalities and complete hyperbolicity of almost complex manifolds. Ann. Inst. Fourier 54, 2387–2435 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kolář, M., Meylan, F.: Infinitesimal CR automorphisms of hypersurfaces of finite type in \({\mathbb{C}}^2\). Arch. Math. (Brno) 47, 367–375 (2011)

    MATH  MathSciNet  Google Scholar 

  30. Lamel, B.: Holomorphic maps of real submanifolds in complex spaces of different dimensions. Pac. J. Math. 201, 357–387 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lamel, B., Mir, N.: Parametrization of local CR automorphisms by finite jets and applications. J. Am. Math. Soc. 20, 519–572 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lee, K.H.: Almost complex manifolds and Cartan’s uniqueness theorem. Trans. Am. Math. Soc. 438, 2057–2069 (2006)

    Article  Google Scholar 

  33. Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. Fr. 109, 427–474 (1981)

    MATH  MathSciNet  Google Scholar 

  34. Lewy, H.: On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables. Ann. Math. 64, 514–522 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  35. Peyron, M.: Analyticité des applications CR dans des variétés presque complexes (preprint). http://arxiv.org/abs/1205.3262

  36. Pinchuk, S.: The scaling method and holomorphic mappings, several complex variables and complex geometry, part 1 (Santa Cruz, CA, 1989). Proc. Sympos. Pure Math. 52 Part 1, Amer. Math. Soc., Providence, pp. 151–161 (1991)

  37. Sato, I.: Almost analytic vector fields in almost complex manifolds. Tohoku Math. J. 17, 185–199 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  38. Sikorav, J.-C.: Some properties of holomorphic curves in almost complex manifolds. In: Audin, M., Lafontaine, J. (eds.) Holomorphic curves in symplectic geometry. Birkhäuser, Basel (1994)

  39. Sukhov, A., Tumanov, A.: Stationary discs and geometry of CR manifolds of codimension two. Int. J. Math. 12, 877–890 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  40. Tanaka, N.: On the pseudo-conformal geometry of hypersurfaces of the space of \(n\) complex variables. J. Math. Soc. Japan 14, 397–429 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  41. Tanaka, N.: On generalized graded Lie algebras and geometric structures. J. Math. Soc. Japan 19, 215–254 (1967) (erratum 36, 1558)

    Google Scholar 

  42. Tumanov, A.: Extremal discs and the regularity of CR mappings in higher codimension. Am. J. Math. 123, 445–473 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  43. Vekua, I.N.: Generalized analytic functions. Addison-Wesley Publishing Company. International series of monographs in pure and applied mathematics (1962)

  44. Zaitsev, D.: Germs of local automorphisms of real-analytic CR structures and analytic dependence on \(k\)-jets. Math. Res. Lett. 4, 823–842 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  45. Zaitsev, D.: Unique determination of local CR-maps by their jets: a survey. Rend. Mat. Acc. Lincei 13, 135–145 (2002)

    Google Scholar 

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Acknowledgments

The authors would like to thank B. Lamel for getting them interested in the problem and for stimulating discussions and helpful remarks.

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Authors and Affiliations

  1. Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, Vienna, 1090, Austria

    Florian Bertrand

  2. Laboratoire Paul Painlevé, Université Lille 1, Villeneuve d’Ascq Cédex, 59655, France

    Léa Blanc-Centi

Authors
  1. Florian Bertrand
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  2. Léa Blanc-Centi
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Corresponding author

Correspondence to Florian Bertrand.

Additional information

Research of F. Bertrand was supported by Austrian Science Fund FWF Grants AY0037721 and M1461-N25.

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Bertrand, F., Blanc-Centi, L. Stationary holomorphic discs and finite jet determination problems. Math. Ann. 358, 477–509 (2014). https://doi.org/10.1007/s00208-013-0972-8

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  • DOI: https://doi.org/10.1007/s00208-013-0972-8

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