Mathematische Zeitschrift

, Volume 274, Issue 1–2, pp 117–144 | Cite as

Some regularity theorems for CR mappings

  • G. P. BalakumarEmail author
  • Kaushal Verma


The purpose of this article is to study Lipschitz CR mappings from an h-extendible (or semi-regular) hypersurface in \({\mathbb{C}^n}\). Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.


CR map Finite type Scaling method Segre varieties 

Mathematics Subject Classification (1991)

Primary 32H40 Secondary 32Q45 32V10 


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  1. 1.
    Baouendi M.S., Rothschild L.P.: Germs of CR maps between real analytic hypersurfaces. Invent. Math. 93, 481–500 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baouendi M.S., Jacobowitz H., Treves F.: On the analyticity of CR mappings. Ann. Math. (2) 122, 365–400 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bedford E.: Proper holomorphic mappings. Bull. Am. Math. Soc. (N.S) 10, 157–175 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bedford E., Barletta E.: Existence of proper mappings from domains in \({\mathbb{C}^2}\). Indiana Univ. Math. J. 39(2), 315–338 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bedford E., Bell S.: Extension of proper holomorphic mappings past the boundary. Manuscripta Math. 50, 1–10 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bedford E., Fornaess J.E.: Local extension of CR functions from weakly pseudoconvex boundaries. Michigan Math. J. 25, 259–262 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bedford E., Pinchuk S.I.: Domains in Cn+1 with noncompact automorphism group. J. Geom. Anal. 1(3), 165–191 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bell, S.: CR maps between hypersurfaces in \({\mathbb{C}^n}\), in Several Complex Variables and Complex Geometry, Part 1 (Santa Cruz, Calif., 1989). In: Proceedings of Symposium on Pure Mathematics, vol. 52, Part 1, pp. 13–22. American Mathematical Society, Providence (1991)Google Scholar
  9. 9.
    Bell S.: Local regularity of CR homeomorphisms. Duke Math. J. 57, 295–300 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bell S., Catlin D.: Regularity of CR mappings. Math. Z. 199, 357–368 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bell S., Narasimhan R.: Proper holomorphic mappings of complex spaces. In: Barth, W., Narasimhan, R. (eds.) Several Complex Variables Encyclopaedia Mathematical Science, vol. 69, Springer, Berlin (1990)Google Scholar
  12. 12.
    Berteloot, F.: Attraction des disques analytiques et continuité höldérienne dpplications holomorphes propres. In: Topics in Complex Analysis (Warsaw, 1992). Banach Center Publ. vol. 31, pp. 91–98 Polish Acad. Sci. Warsaw (1995)Google Scholar
  13. 13.
    Berteloot F.: Principle de Bloch et estimations de la metrique de Kobayashi des domains in \({\mathbb{C}^2}\). J. Geom. Anal. 1, 29–37 (2003)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Berteloot F., Sukhov A.: On the continuous extension of holomorphic correspondences. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24(4), 746–766 (1997)MathSciNetGoogle Scholar
  15. 15.
    Berteloot F., Coeuré G.: Domaines de \({\mathbb{C}^2}\), pseudoconvexes et de type fini ayant un groupe non compact d’automorphismes. Ann. Inst. Fourier (Grenoble) 41(1), 77–86 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Catlin D.: Estimates of invariant metrics on pseudoconvex domains of dimension two. Math. Z. 200, 429–466 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Catlin D.: Boundary invariants of pseudoconvex domains. Ann. Math. (2) 120(3), 529–586 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Cho S.: A lower bound on the Kobayashi metric near a point of finite type in \({\mathbb{C}^n}\). J. Geom. Anal. 2(4), 317–325 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Cho S.: Boundary behavior of the Bergman kernel function on some pseudoconvex domains in \({\mathbb{C}^n}\). Trans. Am. Math. Soc. 345(2), 803–817 (1994)zbMATHGoogle Scholar
  20. 20.
    Coupet B., Pinchuk S.: Holomorphic equivalence problem for weighted homogeneous rigid domains in \({\mathbb{C}^{n+1}}\), Complex analysis in modern mathematics (Russian). pp. 57–70. FAZIS, Moscow (2001)Google Scholar
  21. 21.
    Coupet B., Sukhov A.: On CR mappings between pseudoconvex hypersurfaces of finite type in \({\mathbb{C}^2}\). Duke Math. J. 88(2), 281–304 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Coupet B., Sukhov A.: Reflection principle and boundary properties of holomorphic maps. J. Math. Sci (NY) 125, 825–930 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Coupet B., Gaussier H., Sukhov A.: Regularity of CR maps between convex hypersurfaces of finite type. Proc. Am. Math. Soc. 127(11), 3191–3200 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Coupet B., Pinchuk S., Sukhov A.: On boundary rigidity and regularity of holomorphic mappings. Int. J. Math. 79, 617–643 (1996)CrossRefMathSciNetGoogle Scholar
  25. 25.
    D’Angelo J.: Real hypersurfaces, orders of contact, and applications. Ann. Math. 115, 615–637 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Diederich K., Fornaess J.E.: Pseudoconvex domains: an example with nontrivial Nebenhülle. Math. Ann. 225, 275–292 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Diederich K., Fornaess J.E.: Proper holomorphic images of strictly pseudoconvex domains. Math. Ann. 259(2), 279–286 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Diederich K., Herbort G.: Pseudoconvex domains of semiregular type. In: Contributions to complex analysis and analytic geometry. Aspects of Mathematics, pp. 127–161. Vieweg, Braunschweig (1994)Google Scholar
  29. 29.
    Diederich K., Pinchuk S.: Proper holomorphic maps in dimension 2 extend. Indiana Univ. Math. J. 44, 1089–1126 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Diederich K., Pinchuk S.: Regularity of continuous CR maps in arbitrary dimension. Michigan Math. J. 51, 111–140 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Diederich K., Webster S.: A reflection principle for degenerate real hypersurfaces. Duke Math. J. 47, 835–845 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Fornaess J.E.: Biholomorphic mappings between weakly pseudoconvex domains. Pacific J. Math 74, 63–65 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Fornaess J.E., Sibony N.: Construction of P.S.H. functions on weakly pseudoconvex domains. Duke Math. J. 58(3), 633–655 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Forstneric F.: Proper holomorphic mappings: A survey. In: Fornaess, J.E. (ed.) Several Complex Variables, Proceedings of the special year at the Mittag–Leffler Institute, pp. 297–363. Princeton University Press, Princeton (1993)Google Scholar
  35. 35.
    Gaussier H.: Smoothness of Cauchy Riemann maps for a class of real hypersurfaces. Publ. Mat. 45(1), 79–94 (2001)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Gaussier H.: Tautness and complete hyperbolicity of domains in \({\mathbb{C}^n}\). Proc. Am. Math. Soc. 127(1), 105–116 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Herbort G.: On the invariant differential metrics near pseudoconvex boundary points where the Levi form has corank one. Nagoya Math. J. 130, 25–54 (1993)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Klingenberg W., Pinchuk S.: Normal families of proper holomorphic correspondences. Math. Z. 207 1, 91–96 (1991)CrossRefMathSciNetGoogle Scholar
  39. 39.
    Krantz, S.: Convexity in complex analysis, Several Complex Variables and Complex Geometry, Part 1 (Santa Cruz, Calif., 1989). In: Proceedings of Symposium of Pure Mathematics, vol. 52, Part 1, pp. 119–137. American Mathematical Society, Providence (1991)Google Scholar
  40. 40.
    Lewy H.: On the boundary behaviour of holomorphic mappings. Acad. Naz. Linc. 35, 1–8 (1977)Google Scholar
  41. 41.
    Mahajan, P., Verma, K.: Some aspects of the Kobayashi and Carathéodory metrics on pseudoconvex domains. J. Geom. Anal. (2012) (to appear)Google Scholar
  42. 42.
    Merker J.: On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle. Ann. Inst. Fourier (Grenoble) 52(5), 1443–1523 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Nirenberg L., Webster S., Yang P.: Local boundary regularity of holomorphic mappings. Commun. Pure Appl. Math. 33(3), 305–338 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Pinchuk, S.: On the analytic continuation of holomorphic mappings. Math. USSR Sb. 27, 375–392Google Scholar
  45. 45.
    Pinchuk, S.: Holomorphic inequivalence of certain classes of domains in \({\mathbb{C}^{n}}\). Mat. Sb. (N.S.) 111(153) 1, 67–94, 159 (1980)Google Scholar
  46. 46.
    Pinchuk S., Tsyganov Sh.: Smoothness of CR mappings of strictly pseudoconvex hypersurfaces. Izv. Akad. Nauk. SSSR Ser. Mat. 53, 1120–1129 (1989)Google Scholar
  47. 47.
    Sukhov A.: On boundary behaviour of holomorphic mappings (Russian). Mat. Sb. 185, 131–142 (1994)Google Scholar
  48. 48.
    Sukhov A.: English transl. in Russian Acad. Sci. Sb. Math. 83, 471–483 (1995)CrossRefGoogle Scholar
  49. 49.
    Thai Do D., Thu Ninh V.: Characterization of domains in \({\mathbb{C}^n}\) by their noncompact automorphism groups. Nagoya Math. J. 196, 135–160 (2009)zbMATHMathSciNetGoogle Scholar
  50. 50.
    Trépreau J.M.: Sur le prolongement holomorphe des fonctions CR definies sur une hypersurface reélle de classe C 2 dans \({\mathbb{C}^n}\). Invent. Math. 43, 53–68 (1977)CrossRefMathSciNetGoogle Scholar
  51. 51.
    Verma K.: A Schwarz lemma for correspondences and applications. Publ. Mat. 47(2), 373–387 (2003)zbMATHMathSciNetGoogle Scholar
  52. 52.
    Webster S.: On the mapping problem for algebraic real hypersurfaces. Invent. Math. 43, 53–68 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Yu J.Y.: Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains. Trans. Am. Math. Soc. 347(2), 587–614 (1995)zbMATHCrossRefGoogle Scholar
  54. 54.
    Yu J.Y.: Peak functions on weakly pseudoconvex domains. Indiana Univ. Math. J. 43(4), 1271–1295 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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