Extension of holomorphic maps between real hypersurfaces of different dimension
Annales de l'Institut Fourier, Volume 57 (2007) no. 6, p. 2063-2080

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<nN. 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 CR-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<nN, the boundary of D is real analytic, and the boundary of D is real algebraic, and if f:DD 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.

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<nN. 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 CR 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<nN, la frontière de D est analytique réelle, celle de D’ est algébrique réelle, et si f:DD 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.

DOI : https://doi.org/10.5802/aif.2324
Classification:  32H40
Keywords: Holomorphic mappings, reflection Principle, boundary regularity, analytic continuation
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {6},
     year = {2007},
     pages = {2063-2080},
     doi = {10.5802/aif.2324},
     mrnumber = {2377897},
     zbl = {1149.32008},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_2007__57_6_2063_0/

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