On some rigidity properties of mappings between CR-submanifolds in complex space
Journées équations aux dérivées partielles (2003), article no. 12, 20 p.

We survey some recent results on holomorphic or formal mappings sending real submanifolds in complex space into each other. More specifically, the approximation and convergence properties of formal CR-mappings between real-analytic CR-submanifolds will be discussed, as well as the corresponding questions in the category of real-algebraic CR-submanifolds.

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author = {Meylan, Francine and Mir, Nordine and Zaitsev, Dimitri},
title = {On some rigidity properties of mappings between {CR-submanifolds} in complex space},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
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publisher = {Universit\'e de Nantes},
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Meylan, Francine; Mir, Nordine; Zaitsev, Dimitri. On some rigidity properties of mappings between CR-submanifolds in complex space. Journées équations aux dérivées partielles (2003), article  no. 12, 20 p. doi : 10.5802/jedp.626. http://www.numdam.org/articles/10.5802/jedp.626/

[A68] Artin, M. - On the solutions of analytic equations. Invent. math. 5 (1968), 277-291. | MR 232018 | Zbl 0172.05301

[A69] Artin, M. - Algebraic approximation of structures over complete local rings. Inst. Hautes Etudes Sci. Publ. Math. 36, (1969), 23-58. | Numdam | MR 268188 | Zbl 0181.48802

[BER96] Baouendi, M.S.; Ebenfelt, P.; Rothschild, L.P. - Algebraicity of holomorphic mappings between real algebraic sets in ${C}^{n}$. Acta Math. 177 (1996), 225-273. | MR 1440933 | Zbl 0890.32005

[BER97] Baouendi, M.S.; Ebenfelt, P.; Rothschild, L.P. - Parametrization of local biholomorphisms of real analytic hypersurfaces. Asian J. Math. 1 (1997), 1-16. | MR 1480988 | Zbl 0943.32021

[BER98] Baouendi, M.S.; Ebenfelt, P.; Rothschild, L.P. - CR automorphisms of real analytic manifolds in complex space. Comm. Anal. Geom., 6, (1998), 291-315. | MR 1651418 | Zbl 0982.32030

[BER99a] Baouendi, M.S.; Ebenfelt, P.; Rothschild, L.P. - Real Submanifolds in Complex Space and Their Mappings. Princeton Math. Series 47, Princeton Univ. Press, 1999. | MR 1668103 | Zbl 0944.32040

[BER99b] Baouendi, M.S.; Ebenfelt, P.; Rothschild, L.P. - Rational dependence of smooth and analytic CR mappings on their jets. Math. Ann. 315 (1999), 205-249. | MR 1721797 | Zbl 0942.32027

[BER00a] Baouendi, M.S.; Ebenfelt, P.; Rothschild, L.P. - Convergence and finite determination of formal CR mappings. J. Amer. Math. Soc. 13 (2000), 697-723. | MR 1775734 | Zbl 0958.32033

[BER00b] Baouendi, M.S.; Ebenfelt, P.; Rothschild, L.P. - Local geometric properties of real submanifolds in complex space, Bull. Amer. Math. Soc., 37, (2000) 309-336. | MR 1754643 | Zbl 0955.32027

[BER03] Baouendi, M.S.; Ebenfelt, P.; Rothschild, L.P. - Dynamics of the Segre varieties of a real submanifold in complex space, J. Algebraic Geom., 12 (2003), 81-106. | MR 1948686 | Zbl 1037.32011

[BMR02] Baouendi, M.S.; Mir, N.; Rothschild, L.P. - Reflection ideals and mappings between generic submanifolds in complex space, J. Geom. Anal., 12, no. 4, (2002), 543-580. | MR 1916859 | Zbl 1039.32021

[BRZ01a] Baouendi, M.S.; Rothschild, L.P.; Zaitsev, D. - Equivalences of real submanifolds in complex space. J. Differential Geom. 59 (2001), 301-351. | MR 1908985 | Zbl 1037.32030

[BRZ01b] Baouendi, M.S.; Rothschild, L.P.; Zaitsev, D. - Points in general position in real-analytic submanifolds in ${C}^{N}$ and applications. Ohio State Univ. Math. Res. Inst. Publ., 9, (2001), 1-20. | MR 1912728 | Zbl 1023.32023

[BN90] Bell, S.R.; Narasimhan, R. - Proper holomorphic mappings of complex spaces, Several Complex Variables (Heidelberg) (Barth, W., Narasimhan, R., Eds.), EMS, 69, Springer-Verlag, 1990. | MR 1095089 | Zbl 0733.32021

[BG77] Bloom, T.; Graham, I. - On type conditions for generic real submanifolds of ${C}^{n}$. Invent. math. 40 (1977), 217-243. | MR 589930 | Zbl 0339.32003

[C32a] Cartan, E. - Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, Part I, Ann. Math. Pura Appl. 11, (1932), 17-90. | JFM 58.1256.03

[C32b] Cartan, E. - Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, Part II, Ann. Sc. Norm. Sup. Pisa 1, (1932), 333-354. | JFM 58.1256.03 | Numdam

[CM74] Chern, S.S.; Moser, J.K. - Real hypersurfaces in complex manifolds. Acta Math. 133 (1974), 219-271. | MR 425155 | Zbl 0302.32015

[CMS99] Coupet, B.; Meylan, F.; Sukhov, A. - Holomorphic maps of algebraic CR manifolds. Internat. Math. Res. Notices 1 (1999), 1-29. | MR 1666972 | Zbl 0926.32044

[D'A82] D'Angelo, J.P. - Real hypersurfaces, orders of contact, and applications. Ann. of Math. (2) 115 (1982), no. 3, 615-637. | MR 657241 | Zbl 0488.32008

[D'A91] D'Angelo, J.P. - Finite type and the intersection of real and complex subvarieties. Several complex variables and complex geometry, Part 3(Santa Cruz, CA, 1989), 103-117, Proc. Sympos. Pure Math. 52, Part 3, Amer. Math. Soc., Providence, RI, 1991. | MR 1128587 | Zbl 0745.32008

[Fo93] Forstnerič, F. - Proper holomorphic mappings: a survey. Several complex variables (Stockholm, 1987/1988), 297-363, Math. Notes 38, Princeton Univ. Press, Princeton, NJ, 1993. | MR 1207867 | Zbl 0778.32008

[G02] Gong, X. - Existence of real-analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics, Indiana Univ. Math. J., (to appear). | MR 2048184 | Zbl 1060.32020

[Hu94] Huang, X. - On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions. Ann. Inst. Fourier, (Grenoble) 44, no. 2, 433-463 (1994). | Numdam | MR 1296739 | Zbl 0803.32011

[Hu01] Huang, X. - On some problems in several complex variables and CR geometry. First International Congress of Chinese Mathematicians (Beijing, 1998), 383-396, AMS/IP Stud. Adv. Math. 20, Amer. Math. Soc., Providence, RI, 2001. | MR 1830195 | Zbl 1048.32022

[KZ01] Kim, S.-Y.; Zaitsev, D. - The equivalence and embeddings problems for CR structures of any codimension (preprint 2001).

[K72] Kohn, J.J. - Boundary behavior of ˉ∂ on weakly pseudo-convex manifolds of dimension two. J. Differential Geom. 6, (1972), 523-542. | MR 322365 | Zbl 0256.35060

[La01] Lamel, B. - Holomorphic maps of real submanifolds in complex spaces of different dimensions. Pacific J. Math. 201 (2001), no. 2, 357-387. | MR 1875899 | Zbl 1061.32027

[Le86] Lempert, L. - On the boundary behavior of holomorphic mappings. Contributions to several complex variables, 193-215, Aspects Math., E9, Viehweg, Braunschweig, 1986. | MR 859198 | Zbl 0589.32028

[MMZ03] Meylan, F.; Mir, N.; Zaitsev, D. - Approximation and convergence of formal CR-mappings. Internat. Math. Res. Notices 4, (2003), 211-242. | MR 1935273 | Zbl 1016.32017

[M98] Mir, N. - Germs of holomorphic mappings between real-algebraic hypersurfaces, Ann. Inst. Fourier, (Grenoble) 48 no. 4, (1998), 1025-1043. | Numdam | MR 1656006 | Zbl 0914.32009

[M00] Mir, N.. - Formal biholomorphic maps of real analytic hypersurfaces. Math. Res. Lett. 7 (2000), 343-359. | MR 1764327 | Zbl 0964.32013

[M02a] Mir, N.. - On the convergence of formal mappings. Comm. Anal. Geom. 10, no. 1, (2002), 23-59. | MR 1894140 | Zbl 1008.32022

[M02b] Mir, N.. - Convergence of formal embeddings between real-analytic hypersurfaces in codimension one, J. Differential Geom. 62 (2002), 163-173. | MR 1987381 | Zbl 02171914

[MW83] Moser, J.K.; Webster, S.M. - Normal forms for real surfaces in ${C}^{2}$ near complex tangents and hyperbolic surface transformations. Acta Math. 150 (1983), no. 3-4, 255-296. | MR 709143 | Zbl 0519.32015

[N62] Nagano, T. - Linear differential systems with singularities and an application to transitive Lie algebras. J. Math. Soc. Japan 14, (1962), 397-429. | MR 199865

[Po07] Poincaré, H. - Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo 23, (1907), 185-220. | JFM 38.0459.02

[Pu90a] Pushnikov, A.Yu. - Holomorphicity of CR-mappings into a space of large dimension. Mat. Zametki 48 (1990), no. 3, 147-149. | MR 1088849 | Zbl 0716.32014

[Pu90b] Pushnikov, A.Yu. - On the holomorphy of CR-mappings of real analytic hypersurfaces. Complex analysis and differential equations, 76-84, Bashkir. Gos. Univ., Ufa, 1990. | MR 1252653

[R03] Rothschild, L.P. - Mappings of real submanifolds in complex space, Contemp. Math., Amer. Math. Soc. (to appear). | MR 2018344

[S96] Stanton, N. - Infinitesimal CR automorphisms of real hypersurfaces, Amer. J. Math. 118, (1996), 209-233. | MR 1375306 | Zbl 0849.32012

[Ta62] Tanaka, N. - On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables J. Math. Soc. Japan, 14, 397-429, (1962). | MR 145555 | Zbl 0113.06303

[Tr00] Treves, F. - A treasure trove of geometry and analysis: the hyperquadric, Notices Amer. Math. Soc. 47, no. 10, (2000), 1246 -1256. | MR 1784240 | Zbl 0986.51038

[V85] Vitushkin, A.G. - Holomorphic mappings and the geometry of hypersurfaces, Encyclopedia of Mathematical Sciences, Vol. 7, Several Complex Variables I, Springer-Verlag, Berlin, 1985, pp. 159-214. | MR 850493 | Zbl 0781.32013

[W77] Webster, S. - On the mapping problem for algebraic real hypersurfaces. Invent. math., 43, (1977), 53-68. | MR 463482 | Zbl 0348.32005

[Z97] Zaitsev, D. - Germs of local automorphisms of real analytic CR structures and analytic dependence on k-jets. Math. Res. Lett. 4 (1997), 823-842. | MR 1492123 | Zbl 0898.32006

[Z99] Zaitsev, D. - Algebraicity of local holomorphisms between real-algebraic submanifolds in complex space, Acta Math., 183, (1999), 273-305. | MR 1738046 | Zbl 1005.32014

[Z02] Zaitsev, D. - Unique jet determination of local CR-maps by their jets: a survey, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 13, (2002), 295-305. | MR 1984108 | Zbl 02217222

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