On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

Joël Merker[1]

  • [1] Université de Provence, CMI, 39 rue Joliot-Curie, 13453 Marweille cedex 13 (France)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 5, page 1443-1523
  • ISSN: 0373-0956

Abstract

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In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat “hat”. In our main theorem, we show that every 𝒞 ∞ -smooth CR diffeomorphism h : M → M ' between two globally minimal real analytic hypersurfaces in ℂ n ( n ≥ 2 ) is real analytic at every point of M if M ' is holomorphically nondegenerate. More generally, we establish that the reflection function ℛ h ' associated to such a 𝒞 ∞ -smooth CR diffeomorphism between two globally minimal hypersurfaces in ℂ n ( n ≥ 1 ) always extends holomorphically to a neighborhood of the graph of h ¯ in M × M ¯ ' , without any nondegeneracy condition on M ' . This gives a new version of the Schwarz symmetry principle to several complex variables. Finally, we show that every 𝒞 ∞ - smooth CR mapping h : M → M ' between two real analytic hypersurfaces containing no complex curves is real analytic at every point of M , without any rank condition on h .

How to cite

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Merker, Joël. "On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle." Annales de l’institut Fourier 52.5 (2002): 1443-1523. <http://eudml.org/doc/116015>.

@article{Merker2002,
abstract = {In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat “hat”. In our main theorem, we show that every $\{\mathcal \{C\}\}^\infty $-smooth CR diffeomorphism $h: M\rightarrow M^\{\prime \}$ between two globally minimal real analytic hypersurfaces in $\{\mathbb \{C\}\}^n$ ($n\ge 2$) is real analytic at every point of $M$ if $M^\{\prime \}$ is holomorphically nondegenerate. More generally, we establish that the reflection function $\{\mathcal \{R\}\}_h^\{\prime \}$ associated to such a $\{\mathcal \{C\}\}^\infty $-smooth CR diffeomorphism between two globally minimal hypersurfaces in $\{\mathbb \{C\}\}^n$ ($n\ge 1$) always extends holomorphically to a neighborhood of the graph of $\bar\{h\}$ in $M\times \overline\{M\}^\{\prime \}$, without any nondegeneracy condition on $M^\{\prime \}$. This gives a new version of the Schwarz symmetry principle to several complex variables. Finally, we show that every $\{\mathcal \{C\}\}^\infty $- smooth CR mapping $h: M\rightarrow M^\{\prime \}$ between two real analytic hypersurfaces containing no complex curves is real analytic at every point of $M$, without any rank condition on $h$.},
affiliation = {Université de Provence, CMI, 39 rue Joliot-Curie, 13453 Marweille cedex 13 (France)},
author = {Merker, Joël},
journal = {Annales de l’institut Fourier},
keywords = {reflection principle; continuity principle; CR diffeomorphism; holomorphic nondegeneracy; global minimality in the sense of Trépreau-Tumanov; reflection function; envelopes of holomorphy; envelopes of holomorphy of domains; smooth real analytic hypersurfaces; globally minimal; holomorphically nondegenerate},
language = {eng},
number = {5},
pages = {1443-1523},
publisher = {Association des Annales de l'Institut Fourier},
title = {On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle},
url = {http://eudml.org/doc/116015},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Merker, Joël
TI - On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 5
SP - 1443
EP - 1523
AB - In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat “hat”. In our main theorem, we show that every ${\mathcal {C}}^\infty $-smooth CR diffeomorphism $h: M\rightarrow M^{\prime }$ between two globally minimal real analytic hypersurfaces in ${\mathbb {C}}^n$ ($n\ge 2$) is real analytic at every point of $M$ if $M^{\prime }$ is holomorphically nondegenerate. More generally, we establish that the reflection function ${\mathcal {R}}_h^{\prime }$ associated to such a ${\mathcal {C}}^\infty $-smooth CR diffeomorphism between two globally minimal hypersurfaces in ${\mathbb {C}}^n$ ($n\ge 1$) always extends holomorphically to a neighborhood of the graph of $\bar{h}$ in $M\times \overline{M}^{\prime }$, without any nondegeneracy condition on $M^{\prime }$. This gives a new version of the Schwarz symmetry principle to several complex variables. Finally, we show that every ${\mathcal {C}}^\infty $- smooth CR mapping $h: M\rightarrow M^{\prime }$ between two real analytic hypersurfaces containing no complex curves is real analytic at every point of $M$, without any rank condition on $h$.
LA - eng
KW - reflection principle; continuity principle; CR diffeomorphism; holomorphic nondegeneracy; global minimality in the sense of Trépreau-Tumanov; reflection function; envelopes of holomorphy; envelopes of holomorphy of domains; smooth real analytic hypersurfaces; globally minimal; holomorphically nondegenerate
UR - http://eudml.org/doc/116015
ER -

References

top
  1. R.A. Ajrapetyan, Extension of CR-functions from piecewise smooth CR manifolds, Mat. Sb. 134 (1987), 108-118 Zbl0635.32012
  2. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Algebraicity of holomorphic mappings between real algebraic sets in ℂ n , Acta Math. 177 (1996), 225-273 Zbl0890.32005MR1440933
  3. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Real submanifolds in complex space and their mappings, 47 (1999), Princeton University Press,, Princeton, NJ Zbl0944.32040
  4. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Convergence and finite determinacy of formal CR mappings, J. Amer. Math. Soc. 13 (2000), 697-723 Zbl0958.32033MR1775734
  5. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Local geometric properties of real submanifolds in complex space, Bull. Amer. Math. Soc. 37 (2000), 309-336 Zbl0955.32027MR1754643
  6. M.S. Baouendi, X. Huang, L.P. Rothschild, Regularity of CR mappings between algebraic hypersurfaces, Invent. Math. 125 (1996), 13-36 Zbl0855.32009MR1389959
  7. M.S. Baouendi, H. Jacobowitz, F. Treves, On the analyticity of CR mappings, Ann. of Math. 122 (1985), 365-400 Zbl0583.32021MR808223
  8. M.S. Baouendi, L.P. Rothschild, Germs of CR maps between real analytic hypersurfaces, Invent. Math. 93 (1988), 481-500 Zbl0653.32020MR952280
  9. M.S. Baouendi, L.P. Rothschild, Geometric properties of mappings between hypersurfaces in complex space, J. Differential Geom. 31 (1990), 473-499 Zbl0702.32014MR1037411
  10. M.S. Baouendi, L.P. Rothschild, Cauchy-Riemann functions on manifolds of higher codimension in complex space, Invent. Math. 101 (1990), 45-56 Zbl0712.32009MR1055709
  11. M.S. Baouendi, L.P. Rothschild, Mappings of real algebraic hypersurfaces, J. Amer. Math. Soc. 8 (1995), 997-1015 Zbl0869.14025MR1317227
  12. M.S. Baouendi, F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. 113 (1981), 387-421 Zbl0491.35036MR607899
  13. M.S. Baouendi, F. Treves, About the holomorphic extension of CR functions on real hypersurfaces in complex space, Duke Math. J. 51 (1984), 77-107 Zbl0564.32011MR744289
  14. E. Bedford, J.E. Fornaess, Local extension of CR functions from weakly pseudoconvex boundaries, Michigan Math. J. 25 (1978), 259-262 Zbl0401.32007MR512898
  15. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, E. Bedford, S. Pinchuk, Analytic continuation of biholomorphic maps, Michigan Math. J. 28 (1987), 405-408 Zbl0681.32010
  16. B. Coupet, F. Meylan, A. Sukhov, Holomorphic maps of algebraic CR manifolds, Internat. Math. Res. Notices 1 (1999), 1-29 Zbl0926.32044MR1666972
  17. B. Coupet, S. Pinchuk, A. Sukhov, On partial analyticity of CR mappings, Math. Z. 235 (2000), 541-557 Zbl0972.32008MR1800211
  18. B. Coupet, S. Pinchuk, A. Sukhov, Analyticité des applications CR, C.R. Acad. Sci. Paris, Sér. I Math. 329 (1999), 489-494 Zbl0949.32018MR1715137
  19. J.P. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, (1993), CRC Press, Boca Raton, FL Zbl0854.32001MR1224231
  20. S. Damour, Sur l'algébricité des applications holomorphes, C. R. Acad. Sci. Paris, Sér. I Math. 332 (2001), 491-496 Zbl1066.14509MR1834056
  21. S. Damour, On the analyticity of smooth CR mappings between real analytic CR manifolds, Michigan Math. J. 49 (2001), 583-603 Zbl0997.32033MR1872758
  22. M. Derridj, Le principe de réflexion en des points de faible pseudoconvexité pour des applications holomorphes propres, Invent. Math. 79 (1985), 197-215 Zbl0538.32009MR774535
  23. K. Diederich, J.E. Fornaess, Pseudoconvex domains with real analytic boundary, (1978), 371-384 Zbl0378.32014MR477153
  24. K. Diederich, J.E. Fornaess, Biholomorphic maps between certain real analytic domains in ℂ 2 , Math. Ann. 245 (1979), 255-272 Zbl0418.32018MR553344
  25. K. Diederich, J.E. Fornaess, Biholomorphic mappings between two-dimensional Hartogs domains with real analytic boundaries, Recent Developments in Several Complex Variables 100 (1981), 133-150, Princeton University Press, Princeton N.J. Zbl0504.32011
  26. K. Diederich, J.E. Fornaess, Proper holomorphic mappings between real-analytic pseudoconvex domains in ℂ n ., Math. Ann. 282 (1988), 681-700 Zbl0661.32025MR970228
  27. K. Diederich, J.E. Fornaess, Z. Ye, Biholomorphisms in dimension 2, J. Geom. Anal. 4 (1994), 539-552 Zbl0864.32014MR1305994
  28. K. Diederich, S. Pinchuk, Proper holomorphic maps in dimension 2 extend, Indiana Univ. Math. J. 44 (1995), 1089-1126 Zbl0857.32015MR1386762
  29. K. Diederich, S. Pinchuk, Reflection principle in higher dimensions, ICM, Berlin, Doc. Math. Extra Vol. II (1998), 703-712 Zbl0914.32008
  30. K. Diederich, S.M. Webster, A reflection principle for degenerate real hypersurfaces, Duke Math. J. 47 (1980), 835-843 Zbl0451.32008MR596117
  31. F. Forstneric, Extending proper holomorphic mappings of positive codimension, Invent. Math. 95 (1989), 31-62 Zbl0633.32017MR969413
  32. C.K. Han, Analyticity of CR equivalences between some real analytic hypersurfaces in ℂ n with degenerate Levi-forms, Invent. Math. 73 (1983), 51-69 Zbl0517.32007MR707348
  33. N. Hanges, F. Treves, Propagation of holomorphic extendability of CR functions, Math. Ann. 263 (1983), 157-177 Zbl0494.32004MR698000
  34. X. Huang, Schwarz reflection principle in complex spaces of dimension two, Comm. Partial Differential Equations 21 (1996), 1781-1828 Zbl0886.32010MR1421212
  35. X. Huang, J. Merker, F. Meylan, Mappings between degenerate real analytic hypersurfaces in ℂ n , Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998) 251 (2000), 321-338, Amer. Math. Soc., Providence, RI Zbl0967.32030
  36. B. Jöricke, Deformation of CR-manifolds, minimal points and CR-manifolds with the microlocal analytic extension property, J. Geom. Anal. 6 (1996), 555-611 Zbl0917.32007MR1601405
  37. H. Lewy, On the boundary behaviour of holomorphic mappings, Contrib. Centro Linceo Inter. Sc. Mat. e Loro Appl., Accad. Naz. Lincei 35 (1977) 
  38. H. Maire, F. Meylan, Extension of smooth CR mappings between non-essentially finite hypersurfaces in ℂ 3 , Ark. Math. 35 (1997), 185-199 Zbl0885.32017MR1443041
  39. B. Malgrange, Ideals of Differentiable Functions, No. 3 (1967), Oxford University Press, London Zbl0177.17902
  40. J. Merker, Global minimality of generic manifolds and holomorphic extendibility of CR functions, Internat. Math. Res. Notices (1994), approx. 14 p. (electronic) 329-342 Zbl0815.32007MR1289578
  41. J. Merker, On removable singularities for CR functions in higher codimension, Internat. Math. Res. Notices (1997), approx. 36 p. (electronic) 21-56 Zbl0880.32009MR1426732
  42. J. Merker, On the Schwarz symmetry principle in three-dimensional complex euclidean space, (1997) 
  43. J. Merker, Vector field construction of Segre sets, (1998, augmented in 2000) 
  44. J. Merker, On the partial algebraicity of holomorphic mappings between two real algebraic sets in the complex euclidean spaces of different dimensions, Bull. Soc. Math. France 129 (2001), 547-591 Zbl0998.32019MR1894150
  45. J. Merker, Étude de la régularité analytique de l'application de symétrie CR formelle, (June 2000) 
  46. J. Merker, Convergence of formal invertible CR mappings between minimal holomorphically nondegenerate real analytic hypersurfaces, Int. J. Math. Math. Sci. 26 (2001), 281-302 Zbl1002.32029MR1854009
  47. J. Merker, Étude de la régularité analytique de l'application de symétrie CR formelle, C. R. Acad. Sci. Paris, Sér. I Math. 333 (2001), 165-168 Zbl0994.32028MR1851618
  48. J. Merker, F. Meylan, Extension de germes de difféomorphismes CR pour une classe d’hypersurfaces analytiques réelles non essentiellement finies dans ℂ 3 , Complex variables Theory Appl. 40 (1999), 19-34 Zbl1025.32029MR1742868
  49. J. Merker, F. Meylan, On the Schwarz symmetry principle in a model case, Proc. Amer. Math. Soc. 127 (1999), 1097-1102 Zbl0919.32011MR1476379
  50. J. Merker, E. Porten, On removable singularities for integrable CR functions, Indiana Univ. Math. J. 48 (1999), 805-856 Zbl0935.32010MR1736977
  51. J. Merker, E. Porten, On wedge extendability of CR-meromorphic functions Zbl1026.32020MR1938701
  52. F. Meylan, The reflection principle in complex space, Indiana Univ. Math. J. 44 (1995), 783-796 Zbl0857.32007MR1375349
  53. N. Mir, An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings, Math. Z. 231 (1999), 189-202 Zbl0930.32020MR1696763
  54. N. Mir, Germs of holomorphic mappings between real algebraic hypersurfaces, Ann. Inst. Fourier 48 (1998), 1025-1043 Zbl0914.32009MR1656006
  55. N. Mir, On the convergence of formal mappings Zbl1008.32022MR1894140
  56. N. Mir, Formal biholomorphic maps of real analytic hypersurfaces, Math. Res. Letters 7 (2000), 2-3, 343-359 Zbl0964.32013MR1764327
  57. T.S. Neelon, On solutions of real analytic equations, Proc. Amer. Math. Soc. 125 (1997), 2531-2535 Zbl0890.32004MR1396991
  58. S. Pinchuk, A boundary uniqueness theorem for holomorphic functions of several complex variables, Mat. Zametki 15 (1974), 205-212 Zbl0292.32002MR350065
  59. S. Pinchuk, On proper holomorphic mappings of strictly pseudoconvex domains (Russian), Sibirsk. Mat. Z. 15 (1974), 909-917 Zbl0303.32016
  60. S. Pinchuk, On the analytic continuation of holomorphic mappings (Russian), Mat. Sb. (N.S.) 98(140)-3(11) (1975), 375-392, 416-435, 495-496 Zbl0366.32010MR393562
  61. S. Pinchuk, Holomorphic mappings of real-analytic hypersurfaces (Russian), Mat. Sb. (N.S.) 105(147) (1978), 574-593, 640 Zbl0438.32009MR496595
  62. S. Pinchuk, K. Verma, Analytic sets and the boundary regularity of CR mappings, Proc. Amer. Math. Soc. 129 (2001), 2623-2632 Zbl0981.32024MR1838785
  63. E. Porten, Habilitationsschrift 
  64. C. Rea, Prolongement holomorphe des fonctions CR, conditions suffisantes, C. R. Acad. Sci. Paris, Sér I Math. 297 (1983), 163-165 Zbl0568.32011MR725396
  65. R. Shafikov, Analytic continuation of germs of holomorphic mappings between real hypesurfaces in ℂ n , Michigan Math. J. 47 (2000), 133-149 Zbl0966.32007MR1755261
  66. R. Sharipov, A. Sukhov, On CR mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions, Trans. Amer. Math. Soc. 348 (1996), 767-780 Zbl0851.32017MR1325920
  67. N. Stanton, Infinitesimal CR automorphisms of rigid hypersurfaces of the space of n complex variables, Amer. J. Math. 117 (1995), 141-167 Zbl0826.32013MR1314461
  68. N. Stanton, Infinitesimal CR automorphisms of real hypersurfaces, Amer. J. Math. 118 (1996), 209-233 Zbl0849.32012MR1375306
  69. A. Sukhov, On the mapping problem for quadric Cauchy-Riemann manifolds, Indiana Univ. Math. J. 42 (1993), 27-36 Zbl0848.32016MR1218705
  70. A. Sukhov, On CR mappings of real quadric manifolds, Michigan Math. J. 41 (1994), 143-150 Zbl0821.32015MR1260615
  71. J.-M. Trépreau, Sur le prolongement holomorphe des fonctions CR définies sur une hypersurface réelle de classe 𝒞 2 dans ℂ n , Invent. Math. 83 (1986), 583-592 Zbl0586.32016MR827369
  72. J.-M. Trépreau, Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. Fr. 118 (1990), 403-450 Zbl0742.58053MR1090408
  73. J.-M. Trépreau, Holomorphic extension of CR functions: a survey, Partial differential equations and mathematical physics (Copenhagen, 1995; Lund 1995) 21 (1996), 333-355, Birkhäuser Boston, Boston, MA Zbl0851.32021
  74. A.E. Tumanov, Extending CR functions on a manifold of finite type over a wedge (Russian), Mat. Sb. (N.S.) 136(178) (1988), 128-139 Zbl0692.58005MR945904
  75. A.E. Tumanov, Connections and propagation of analyticity for CR functions, Duke Math. J. 73 (1994), 1-24 Zbl0801.32005MR1257276
  76. A.E. Tumanov, On the propagation of extendibility of CR functions, Complex analysis and geometry (Trento, 1993) 173 (1996), 479-498, Dekker, New York Zbl0849.32013
  77. K. Verma, Boundary regularity of correspondences in ℂ 2 , Math. Z. 231 (1999), 253-299 Zbl0939.32014MR1703349
  78. S.M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977), 53-68 Zbl0348.32005MR463482
  79. S.M. Webster, On the reflection principle in several complex variables, Proc. Amer. Math. Soc. 71 (1978), 26-28 Zbl0626.32019MR477138
  80. S.M. Webster, Holomorphic mappings of domains with generic corners, Proc. Amer. Math. Soc. 86 (1982), 236-240 Zbl0505.32014MR667281
  81. M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277-291 Zbl0172.05301MR232018
  82. R.A. Ajrapetyan, Extension of CR-functions from piecewise smooth CR manifolds, Math. USSR Sb. (English transl.) 62 (1989), 111-120 Zbl0663.32015MR912414
  83. A.E. Tumanov, Extending CR functions on a manifold of finite type over a wedge, Math. USSR Sb. 64 (1989), 129-140 Zbl0692.58005MR945904

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