×

zbMATH — the first resource for mathematics

Mappings of quadric Cauchy-Riemann manifolds. (English) Zbl 0749.32006
On considère les variétés \(M\) dans \(\mathbb{C}^{m+d}\), dites de type \((d,m)\) (\(d\) et \(m>0)\), définies par d’équations \(\text{Im} w_ k=\langle A_ kz,\overline z\rangle\) dont les seconds membres sont des formes hermitiennes sur \(\mathbb{C}^ m\); à une telle \(M\) est associé dans \(\mathbb{R}^ d\) le cône enveloppe convexe de l’ensemble décrit par le point de coordonnées \(\langle A_ kz,\overline z\rangle\), \(z\in\mathbb{C}^ m\); on dit que \(M\) est Levi-nondégénérée si \(A_ kz=0\) \(\forall k\) équivaut à \(z=0\). Soient alors \(M\), de type \((d,m)\), ayant un cône associé d’intérieur nonvide et \(M'\), de type \((d',m)\), Levi-nondégénérée, \(\omega\) une partie ouverte et connexe de \(M\): si \(F\in{\mathcal C}^ 1(\omega,M')\) satisfait sur \(\omega\) aux équations de Cauchy-Riemann tangentielles et si, pour un point \(p\in\omega\), \(dF(p)\) est un isomorphisme de l’espace tangent complexe \(T_ pM\) (de dimension \(m)\) sur \(T_{F(p)}M'\), alors \(F\) se prolonge à \(\mathbb{C}^{m+d}\) en une application rationnelle, dont le degré est majoré par un nombre ne dépendant que du type \((d,m)\) de \(M\). Si \(M'\) est de type \((d',m')\), ce prolongement subsiste sous deux hypothèses nouvelles: le cône associé à \(M'\), diminué de l’origine, est contenu dans un demi-espace ouvert de \(\mathbb{R}^ d\); sur un voisinage de \(p\) dans \(\omega\), \(F\) est de classe \({\mathcal C}^ s\), où \(s=m'-r(p)+1\) et \(r(p)\) est le rang de \(dF(p):T_ pM\to T_{F(p)}M'\).
Reviewer: M.Hervé (Paris)

MSC:
32D15 Continuation of analytic objects in several complex variables
32V40 Real submanifolds in complex manifolds
32H35 Proper holomorphic mappings, finiteness theorems
32H40 Boundary regularity of mappings in several complex variables
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Alexander, H.: Holomorphic mappings from the ball and polydisc. Math. Ann.209, 249–256 (1974) · Zbl 0281.32019 · doi:10.1007/BF01351851
[2] Baouendi, M.S., Bell, S., Rothschild, L.P.: Mappings of three-dimensional CR manifolds and their holomorphic extension. Duke Math. J.56, 503–530 (1988) · Zbl 0655.32015 · doi:10.1215/S0012-7094-88-05621-9
[3] Baouendi, M.S., Chang, C.H., Treves, F.: Microlocal hypo-ellipticity and extension of CR functions. J. Differ. Geom.18, 331–391 (1983) · Zbl 0575.32019
[4] Baouendi, M.S., Jacobowitz, H., Treves, F.: On the analyticity of CR mappings. Ann. Math.122, 365–400 (1985) · Zbl 0583.32021 · doi:10.2307/1971307
[5] Baouendi, M.S., Rothschild, L.P.: Germs of CR maps between analytic real hypersurfaces. Invent. Math.93, 481–500 (1988) · Zbl 0653.32020 · doi:10.1007/BF01410197
[6] Baouendi, M.S., Rothschild, L.P.: A general reflection principle inC 2. (Preprint 1989)
[7] Baouendi, M.S., Rothschild, L.P.: Normal forms for generic manifolds and holomorphic extension of CR functions. J. Differ. Geom.25, 431–467 (1987) · Zbl 0629.32016
[8] Baouendi, S., Treves, F.: A property of the functions and distributions annihilated by a locally integrable system of complex vectorfields. Ann. Math.113, 387–421 (1981) · Zbl 0491.35036 · doi:10.2307/2006990
[9] Boggess, A., Polking, J.C.: Holomorphic extension of CR functions. Duke Math. J.49, 757–784 (1982) · Zbl 0506.32003 · doi:10.1215/S0012-7094-82-04938-9
[10] Bochner, S., Martin, W.T.: Several complex variables. Princeton: Princeton University Press 1948 · Zbl 0041.05205
[11] Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math.133, 219–271 (1975) · Zbl 0302.32015 · doi:10.1007/BF02392146
[12] Cima, J., Suffridge, T.J.: A reflection principle with applications to proper holomorphic mappings. Math. Ann.265, 489–500 (1983) · Zbl 0525.32021 · doi:10.1007/BF01455949
[13] Coupet, B.: Construction de disques analytiques et applications. C.R. Acad. Sci., Paris, Sér. I304 (no. 14), 427–430 (1987) · Zbl 0617.32008
[14] Coupet, B.: Régularité d’applications holomorphes sur des variétés totalement réelles; Structure des espaces de Bergman. Thèse, Universite de Provence, Marseille, 1987
[15] D’Angelo, J.: Proper holomorphic maps between balls of different dimensions. Mich. Math. J.35, 83–90 (1988) · Zbl 0651.32014 · doi:10.1307/mmj/1029003683
[16] D’Angelo, J.: Polynomial proper maps between balls. Duke Math. J.57, 211–219 (1988) · Zbl 0657.32012 · doi:10.1215/S0012-7094-88-05710-9
[17] D’Angelo, J.: The structure of proper rational holomorphic maps between balls. (Preprint 1988)
[18] Diederich, K., Fornæss, J.E.: Proper holomorphic mappings between real-analytic pseudoconvex domains inC n . Math. Ann.282, 681–700 (1988) · Zbl 0661.32025 · doi:10.1007/BF01462892
[19] Diederich, K., Fornæss, J.E.: Applications holomorphes propres entre domaines à bord analytique réel. C.R. Acad. Sci. Paris, Sér. I307, 321–324 (1988) · Zbl 0656.32013
[20] Diederich, K., Webster, S.: A reflection principles for degenerate real hypersurfaces. Duke Math. J.47, 835–843 (1980) · Zbl 0451.32008 · doi:10.1215/S0012-7094-80-04749-3
[21] Dor, A.: Proper holomorphic maps from strongly pseudoconvex domains inC 2 to the unit ball inC 3 and boundary interpolation by proper holomorphic maps. (Preprint 1987)
[22] Forstnerič, F.: Extending proper holomorphic mappings of positive codimension. Invent. Math.95, 31–62 (1989) · Zbl 0633.32017 · doi:10.1007/BF01394144
[23] Hakim, M.: Applications holomorphes propres continues de domaines strictement pseudoconvexes deC n dans la boule unitéC n+1 . (Preprint 1987)
[24] Henkin, G.M., Novikov, R.: Proper mappings of classical domains. In: Linear and complex analysis problem book, pp. 625–627. Berlin Heidelberg New York: Springer 1984
[25] Lewy, H.: On the boundary behavior of holomorphic mappings. Atti Accad. Naz. Lincei35, 1–8 (1977) · Zbl 0377.31008
[26] Naruki, I.: Holomorphic extension problem for standard real submanifolds of second level. Publ. Res. Inst. Math. Sci.6, 113–187 (1970) · Zbl 0225.32008 · doi:10.2977/prims/1195194189
[27] Piatetsky-Shapiro, I.I.: Géometrie des domaines classiques et théorie des fonctions automorphes. Paris: Dunod 1966
[28] Pinčuk, S.I.: On the analytic continuation of biholomorphic mappings (in Russian). Mat. Sb.98 (18), 416–435 (1975); English transl. in Math. USSR, Sb.27, 375–392 (1975)
[29] Pinčuk, S.I., Hasanov, S.V.: Asymptotically holomorphic functions (in Russian). Mat. Sb.134 (176), 546–555 (1987)
[30] Pinčuk, S.I., Tsyganov, Sh.I.: Smoothness of CR mappings of strongly pseudoconvex hypersurfaces (in Russian). Izv. Akad. Nauk SSSR53, 1120–1129 (1989)
[31] Rudin, W.: Function theory on the unit ball ofC n . New York: Springer 1980 · Zbl 0495.32001
[32] Sadullaev, A.: A boundary uniqueness theorem inC n (in Russian). Mat. Sb.101, 501–514 (1976) · Zbl 0385.32007 · doi:10.1070/SM1976v030n04ABEH002285
[33] Segre, B.: Intorno al problem di Poincaré della representazione pseudo-conform. Rend. Atti Accad. Naz. Lincei, VI. Ser.13, 676–683 (1931) · Zbl 0003.21302
[34] Tumanov, A.E.: Extending CR functions to a wedge from manifolds of finite type (in Russian). Mat. Sb.136 (178), 128–139 (1988)
[35] Tumanov, A.E.: Finite dimensionality of the group of C-R automorphisms of standard C-R manifolds and proper holomorphic mappings of Siegel domains (in Russian) Izv. Akad. Nauk SSSR52, 651–659 (1988) · Zbl 0655.32026
[36] Tumanov, A.E., Henkin, G.M.: Local characterization of holomorphic automorphisms of Siegel domains (in Russian) Funkts. Anal.17, 49–61 (1983)
[37] Tumanov, A.E., Henkin, G.M.: Local characterization of holomorphic automorphisms of classical domains (in Russian). Dokl. Akad. Nauk SSSR267, 796–799 (1982)
[38] Webster, S.M.: On the mapping problem for algebraic real hypersurfaces. Invent. Math.43, 53–68 (1977) · Zbl 0355.32026 · doi:10.1007/BF01390203
[39] Webster, S.M.: Holomorphic mappings of domains with generic corners. Proc. Am. Math. Soc.86, 236–240 (1982) · Zbl 0505.32014 · doi:10.1090/S0002-9939-1982-0667281-X
[40] Webster, S.M.: Analytic discs and the regularity of C-R mappings of real submanifolds inC n Proc. Symp. Pure Math.41, 199–208 (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.