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 be a connected smooth real analytic minimal hypersurface in , be a compact strictly pseudoconvex real algebraic hypersurface in , . Suppose that is a germ of a holomorphic map at a point in and is in . Then f extends as a holomorphic map along any smooth -curve on M with the extension sending to . Further, if and are smoothly bounded domains in and respectively, , the boundary of is real analytic, and the boundary of is real algebraic, and if is a proper holomorphic map which admits a smooth extension to a neighbourhood of a point in the boundary of , then the map extends continuously to the closure of , and the extension is holomorphic on a dense open subset of the boundary of .
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 une hypersurface lisse, connexe, analytique réelle et minimale dans , et une hypersurface compacte, strictement pseudoconvexe, et algébrique réelle dans , avec . Supposons que soit le germe d’une application holomorphe en un point de , et soit contenu dans . Alors se prolonge à un application holomorphe le long de toute courbe sur , et le prolongement envoie dans . De plus, si et sont des domaines bornés lisses dans et respectivement, avec , la frontière de est analytique réelle, celle de D’ est algébrique réelle, et si est une application holomorphe propre qui admet un prolongement lisse à un voisinage d’un point de la frontière de , alors l’application se prolonge continûment à la fermeture de , et le prolongement est analytique sur un sous-ensemble dense de la frontière de .
Keywords: Holomorphic mappings, reflection Principle, boundary regularity, analytic continuation
Keywords: applications holomorphes, principe de réflexion, prolongement analytique
Shafikov, Rasul 1; Verma, Kausha 2
@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}, mrnumber = {2377897}, zbl = {1149.32008}, language = {en}, url = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Volume 57 (2007) no. 6, pp. 2063-2080. doi : 10.5802/aif.2324. https://aif.centre-mersenne.org/articles/10.5802/aif.2324/
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