Extension of holomorphic maps between real hypersurfaces of different dimension

Shafikov, Rasul; Verma, Kausha

doi:10.5802/aif.2324

Shafikov, Rasul ¹ ; Verma, Kausha ²

¹ University of Western Ontario Department of Mathematics London N6A 5B7 (Canada)
² Indian Institute of Science Department of Mathematics Bangalore 560012 (India)

Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 2063-2080.

Résumé
Abstract

Dans cet article, les résultats sur le prolongement analytique des germes d’applications holomorphes d’une hypersurface analytique réelle à une hypersurface algébrique réelle sont étendus au cas où la cible est une hypersurface de dimension supérieure à celle de la source. Plus précisément, nous prouvons ce qui suit : soit $M$ une hypersurface lisse, connexe, analytique réelle et minimale dans $C^{n}$ , et $M^{'}$ une hypersurface compacte, strictement pseudoconvexe, et algébrique réelle dans $C^{N}$ , avec $1 < n \leq N$ . Supposons que $f$ soit le germe d’une application holomorphe en un point $p$ de $M$ , et $f (M)$ soit contenu dans $M^{'}$ . Alors $f$ se prolonge à un application holomorphe le long de toute courbe $C R$ sur $M$ , et le prolongement envoie $M$ dans $M^{'}$ . De plus, si $D$ et $D^{'}$ sont des domaines bornés lisses dans $C^{n}$ et $C^{N}$ respectivement, avec $1 < n \leq N$ , la frontière de $D$ est analytique réelle, celle de D’ est algébrique réelle, et si $f : D \to D^{'}$ est une application holomorphe propre qui admet un prolongement lisse à un voisinage d’un point $p$ de la frontière de $D$ , alors l’application $f$ se prolonge continûment à la fermeture de $D$ , et le prolongement est analytique sur un sous-ensemble dense de la frontière de $D$ .

In this paper we extend the results on analytic continuation of germs of holomorphic mappings from a real analytic hypersurface to a real algebraic hypersurface to the case when the target hypersurface is of higher dimension than the source. More precisely, we prove the following: Let $M$ be a connected smooth real analytic minimal hypersurface in $C^{n}$ , $M^{'}$ be a compact strictly pseudoconvex real algebraic hypersurface in $C^{N}$ , $1 < n \leq N$ . Suppose that $f$ is a germ of a holomorphic map at a point $p$ in $M$ and $f (M)$ is in $M^{'}$ . Then f extends as a holomorphic map along any smooth $C R$ -curve on M with the extension sending $M$ to $M^{'}$ . Further, if $D$ and $D^{'}$ are smoothly bounded domains in $C^{n}$ and $C^{N}$ respectively, $1 < n \leq N$ , the boundary of $D$ is real analytic, and the boundary of $D^{'}$ is real algebraic, and if $f : D \to D^{'}$ is a proper holomorphic map which admits a smooth extension to a neighbourhood of a point $p$ in the boundary of $D$ , then the map $f$ extends continuously to the closure of $D$ , and the extension is holomorphic on a dense open subset of the boundary of $D$ .

EuDML MR Zbl

DOI : 10.5802/aif.2324

Classification : 32H40
Keywords: Holomorphic mappings, reflection Principle, boundary regularity, analytic continuation
Mots-clés : applications holomorphes, principe de réflexion, prolongement analytique

Affiliations des auteurs :

Shafikov, Rasul ¹ ; Verma, Kausha ²

¹ University of Western Ontario Department of Mathematics London N6A 5B7 (Canada)
² Indian Institute of Science Department of Mathematics Bangalore 560012 (India)

@article{AIF_2007__57_6_2063_0,
     author = {Shafikov, Rasul and Verma, Kausha},
     title = {Extension of holomorphic maps between real hypersurfaces of different dimension},
     journal = {Annales de l'Institut Fourier},
     pages = {2063--2080},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {6},
     year = {2007},
     doi = {10.5802/aif.2324},
     zbl = {1149.32008},
     mrnumber = {2377897},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2324/}
}

TY  - JOUR
AU  - Shafikov, Rasul
AU  - Verma, Kausha
TI  - Extension of holomorphic maps between real hypersurfaces of different dimension
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 2063
EP  - 2080
VL  - 57
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2324/
DO  - 10.5802/aif.2324
LA  - en
ID  - AIF_2007__57_6_2063_0
ER  -

%0 Journal Article
%A Shafikov, Rasul
%A Verma, Kausha
%T Extension of holomorphic maps between real hypersurfaces of different dimension
%J Annales de l'Institut Fourier
%D 2007
%P 2063-2080
%V 57
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2324/
%R 10.5802/aif.2324
%G en
%F AIF_2007__57_6_2063_0

Shafikov, Rasul; Verma, Kausha. Extension of holomorphic maps between real hypersurfaces of different dimension. Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 2063-2080. doi : 10.5802/aif.2324. http://www.numdam.org/articles/10.5802/aif.2324/

Bibliographie
Cité par

[1] Alexander, H. Holomorphic mappings from the ball and polydisc, Math. Ann., Volume 209 (1974), pp. 249-256 | DOI | MR | Zbl

[2] Baouendi, M. S.; Ebenfelt, P.; Rothschild, L. P. Algebraicity of holomorphic mappings between real algebraic sets in $C^{n}$ , Acta Math., Volume 177 (1996) no. 2, pp. 225-273 | DOI | MR | Zbl

[3] Baouendi, M. Salah; Ebenfelt, Peter; Rothschild, Linda Preiss Real submanifolds in complex space and their mappings, Princeton Mathematical Series, 47, Princeton University Press, Princeton, NJ, 1999 | MR | Zbl

[4] Chirka, E. M. Complex analytic sets, Mathematics and its Applications (Soviet Series), 46, Kluwer Academic Publishers Group, Dordrecht, 1989 (Translated from the Russian by R. A. M. Hoksbergen) | MR | Zbl

[5] Coupet, Bernard; Damour, Sylvain; Merker, Joël; Sukhov, Alexandre Sur l’analyticité des applications CR lisses à valeurs dans un ensemble algébrique réel, C. R. Math. Acad. Sci. Paris, Volume 334 (2002) no. 11, pp. 953-956 | Zbl

[6] Coupet, Bernard; Meylan, Francine; Sukhov, Alexandre Holomorphic maps of algebraic CR manifolds, Internat. Math. Res. Notices (1999) no. 1, pp. 1-29 | DOI | MR | Zbl

[7] Diederich, K.; Fornæss, J. E. Pseudoconvex domains with real-analytic boundary, Ann. Math. (2), Volume 107 (1978) no. 2, pp. 371-384 | DOI | MR | Zbl

[8] Diederich, K.; Fornæss, J. E. Proper holomorphic mappings between real-analytic pseudoconvex domains in $C^{n}$ , Math. Ann., Volume 282 (1988) no. 4, pp. 681-700 | DOI | MR | Zbl

[9] Diederich, K.; Sukhov, A. Extension of CR maps into hermitian quadrics (2005) (preprint)

[10] Diederich, K.; Webster, S. M. A reflection principle for degenerate real hypersurfaces, Duke Math. J., Volume 47 (1980) no. 4, pp. 835-843 | DOI | MR | Zbl

[11] Dor, Avner Proper holomorphic maps between balls in one co-dimension, Ark. Mat., Volume 28 (1990) no. 1, pp. 49-100 | DOI | MR | Zbl

[12] Forstnerič, Franc Embedding strictly pseudoconvex domains into balls, Trans. Amer. Math. Soc., Volume 295 (1986) no. 1, pp. 347-368 | MR | Zbl

[13] Forstnerič, Franc Extending proper holomorphic mappings of positive codimension, Invent. Math., Volume 95 (1989) no. 1, pp. 31-61 | DOI | MR | Zbl

[14] Globevnik, Josip Boundary interpolation by proper holomorphic maps, Math. Z., Volume 194 (1987) no. 3, pp. 365-373 | DOI | MR | Zbl

[15] Hakim, Monique Applications holomorphes propres continues de domaines strictement pseudoconvexes de $C^{n}$ dans la boule unité de $C^{n + 1}$ , Duke Math. J., Volume 60 (1990) no. 1, pp. 115-133 | DOI | MR | Zbl

[16] Huang, Xiao Jun On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions, Ann. Inst. Fourier (Grenoble), Volume 44 (1994) no. 2, pp. 433-463 | DOI | Numdam | MR | Zbl

[17] Łojasiewicz, Stanisław Introduction to complex analytic geometry, Birkhäuser Verlag, Basel, 1991 (Translated from the Polish by Maciej Klimek) | Zbl

[18] Løw, Erik Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls, Math. Z., Volume 190 (1985) no. 3, pp. 401-410 | DOI | MR | Zbl

[19] Merker, Joël On the partial algebraicity of holomorphic mappings between two real algebraic sets, Bull. Soc. Math. France, Volume 129 (2001) no. 4, pp. 547-591 | Numdam | MR | Zbl

[20] Merker, Joël; Porten, Egmont On wedge extendability of CR-meromorphic functions, Math. Z., Volume 241 (2002) no. 3, pp. 485-512 | DOI | MR | Zbl

[21] Merker, Joël; Porten, Egmont Holomorphic extension of CR functions, envelopes of holomorphy, and removable singularities, IMRS Int. Math. Res. Surv. (2006), pp. 1-287 | MR | Zbl

[22] Meylan, Francine; Mir, Nordine; Zaitsev, Dmitri Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds, Asian J. Math., Volume 7 (2003) no. 4, pp. 493-509 | MR | Zbl

[23] Nemirovskiĭ; Shafikov, R. G. Uniformization of strictly pseudoconvex domains, I, II, Izv. Ross. Akad. Nauk Ser. Mat., Volume 69 (2005) no. 6, p. 1189-1202, 1203–1210 | MR | Zbl

[24] Pinchuk, S. I. On holomorphic maps of real-analytic hypersurfaces, Math. USSR Sb., Volume 34 (1978), pp. 503-519 | DOI | Zbl

[25] Pinchuk, S. I. Analytic continuation of holomorphic mappings and the problem of holomorphic classification of multidimensional domains, Moscow State Univ. (1980) Doctoral dissertation (Habilitation)

[26] Pinchuk, S. I.; Khenkin, G. M. Holomorphic maps in $C^{n}$ and the problem of holomorphic equivalence, Encyclopaedia of Mathematical Sciences: Several Complex Variables III, Volume 9, Springer-Verlag, 1989 | Zbl

[27] Pinchuk, S. I.; Sukhov, A. Extension of CR maps of positive codimension, Proc. Steklov Inst. Math., Volume 253 (2006), pp. 246-255 | DOI | MR

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

[29] Shafikov, Rasul Analytic continuation of germs of holomorphic mappings between real hypersurfaces in $C^{n}$ , Michigan Math. J., Volume 47 (2000) no. 1, pp. 133-149 | DOI | MR | Zbl

[30] Shafikov, Rasul On boundary regularity of proper holomorphic mappings, Math. Z., Volume 242 (2002) no. 3, pp. 517-528 | DOI | MR | Zbl

[31] Shafikov, Rasul Analytic continuation of holomorphic correspondences and equivalence of domains in $ℂ^{n}$ , Invent. Math., Volume 152 (2003) no. 3, pp. 665-682 | DOI | MR | Zbl

[32] Sharipov, Ruslan; Sukhov, Alexander On CR-mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions, Trans. Amer. Math. Soc., Volume 348 (1996) no. 2, pp. 767-780 | DOI | MR | Zbl

[33] Tanaka, Noboru On the pseudo-conformal geometry of hypersurfaces of the space of $n$ complex variables, J. Math. Soc. Japan, Volume 14 (1962), pp. 397-429 | DOI | MR | Zbl

[34] Vitushkin, A. G. Real-analytic hypersurfaces of complex manifolds, Russian Math. Surveys, Volume 40 (1985), pp. 1-35 | DOI | MR | Zbl

[35] Webster, S. M. On the mapping problem for algebraic real hypersurfaces, Invent. Math., Volume 43 (1977) no. 1, pp. 53-68 | DOI | MR | Zbl

[36] Zaitsev, Dmitri Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces, Acta Math., Volume 183 (1999) no. 2, pp. 273-305 | DOI | MR | Zbl

Cité par Sources :