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ON BOUNDARY REGULARITY OF HOLOMORPHIC CORRESPONDENCES

  • Ourimi, Nabil (Department of Mathematics King Saud University)
  • Received : 2010.07.19
  • Published : 2012.01.01

Abstract

Let D be an arbitrary domain in $\mathbb{C}^n$, n > 1, and $M{\subset}{\partial}D$ be an open piece of the boundary. Suppose that M is connected and ${\partial}D$ is smooth real-analytic of finite type (in the sense of D'Angelo) in a neighborhood of $\bar{M}$. Let f : $D{\rightarrow}\mathbb{C}^n$ be a holomorphic correspondence such that the cluster set $cl_f$(M) is contained in a smooth closed real-algebraic hypersurface M' in $\mathbb{C}^n$ of finite type. It is shown that if f extends continuously to some open peace of M, then f extends as a holomorphic correspondence across M. As an application, we prove that any proper holomorphic correspondence from a bounded domain D in $\mathbb{C}^n$ with smooth real-analytic boundary onto a bounded domain D' in $\mathbb{C}^n$ with smooth real-algebraic boundary extends as a holomorphic correspondence to a neighborhood of $\bar{D}$.

Keywords

analytic sets;holomorphic correspondences;Segre varieties

Acknowledgement

Supported by : King Saud University

References

  1. B. Ayed and N. Ourimi, Analytic continuation of holomorphic mappings, C. R. Math. Acad. Sci. Paris 347 (2009), no. 17-18, 1011-1016. https://doi.org/10.1016/j.crma.2009.07.001
  2. B. Ayed and N. Ourimi, A Local extension of proper holomorphic maps between some unbounded domains in $C^{n}$, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 3, 513-534.
  3. M. S. Baouendi and L. P. Rotshschild, Germs of CR maps between real analytic hyper-surfaces, Invent. Math. 93 (1998), no. 3, 481-500.
  4. M. S. Baouendi and L. P. Rotshschild, Images of real hypersurfaces under holomorphic mappings, J. Differential Geom. 36 (1992), no. 1, 75-88. https://doi.org/10.4310/jdg/1214448443
  5. F. Berteloot and A. Sukhov, On the continuous extension of holomorphic correspon-dences, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 4, 747-766.
  6. E. M. Chirka, Complex Analytic Sets, Kluwer Academic Publishers, 1989.
  7. S. Chern and J. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. https://doi.org/10.1007/BF02392146
  8. K. Diederich and J. E. Fornaess, Proper holomorphic mappings between real-analytic pseudoconvex domains in $C^n$, Math. Ann. 282 (1988), no. 4, 681-700. https://doi.org/10.1007/BF01462892
  9. K. Diederich and S. Pinchuk, Analytic sets extending the graphs of holomorphic mappings, J. Geom. Anal. 14 (2004), no. 2, 231-239. https://doi.org/10.1007/BF02922070
  10. K. Diederich and S. Pinchuk, Proper holomorphic maps in dimension 2 extend, Indiana Univ. Math. J. 44 (1995), no. 4, 1089-1126.
  11. K. Diederich and S. Pinchuk, Regularity of continuous CR maps in arbitrary dimension, Mich. Math. J. 51 (2003), no. 1, 111-140. https://doi.org/10.1307/mmj/1049832896
  12. K. Diederich and J. Fornaess, Proper holomphic mapping between real-analytic pseudo- convex domains in , Math. Ann. 282 (1988), no. 4, 681-700. https://doi.org/10.1007/BF01462892
  13. K. Diederich and S. Webster, A reflection principle for degenerate real hypersurfaces, Duke Math. J. 47 (1980), no. 4, 835-845. https://doi.org/10.1215/S0012-7094-80-04749-3
  14. J. Merker and E. Porten, On wedge extendability of CR-meromorphic functions, Math. Z. 241 (2002), no. 3, 485-512. https://doi.org/10.1007/s00209-002-0426-6
  15. S. Pinchuk, On holomorphic maps of real-analytic hypersurfaces, Math. USSR Sb. 34 (1978), 503-519. https://doi.org/10.1070/SM1978v034n04ABEH001224
  16. S. Pinchuk, Analytic continuation of holomorphic mappings and the problem of holomorphic classification of multidimensional domains, Doctoral dissertation (habilitation), Moscow State Univ., 1980.
  17. S. Pinchuk and K. Verma, Analytic sets and the boundary regularity of CR mappings, Proc. Amer. Math. Soc. 129 (2001), no. 9, 2623-2632. https://doi.org/10.1090/S0002-9939-01-05970-6
  18. R. Shafikov, Analytic continuation of germs of holomorphic mappings, Mich. Math. J. 47 (2000), no. 1, 133-149. https://doi.org/10.1307/mmj/1030374673
  19. R. Shafikov, Analytic continuation of holomorphic correspondences and equivalence of domains in $C^n$, Invent. Math. 152 (2003), no. 3, 665-682. https://doi.org/10.1007/s00222-002-0282-3
  20. R. Shafikov, On boundary regularity of proper holomorphic mappings, Math. Z. 242 (2002), no. 3, 517-528. https://doi.org/10.1007/s002090100355
  21. R. Shafikov and K. Verma, A local extension theorem for proper holomorphic mappings in $C^2$, J. Geom. Anal. 13 (2003), no. 4, 697-714. https://doi.org/10.1007/BF02921885
  22. R. Shafikov and K. Verma, Extension of holomorphic maps between real hypersurfaces of different dimension, Ann. Inst. Fourier, Grenoble 57 (2007), no. 6, 2063-2080. https://doi.org/10.5802/aif.2324
  23. R. Shafikov and K. Verma, Boundary regularity of correspondences in $C^n$, Proc. Indian Acad. Sci. (Math. Sci.) 116 (2006), no. 1, 59-70. https://doi.org/10.1007/BF02829739
  24. K. Verma, Boundary regularity of correspondences in $C^2$, Math. Z. 231 (1999), no. 2, 253-299. https://doi.org/10.1007/PL00004728
  25. S. M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math 43 (1977), no. 1, 53-68. https://doi.org/10.1007/BF01390203