Geodesic
Levi and Tr\'epreau Theorems for Continuous Graphs
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 235 (2001), pp. 272-287.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\Gamma\subset\mathbb C^{n+1}$ be a continuous graph over a convex domain $D\subset\mathbb C^n\times\mathbb R$ and $a\in\Gamma$ be a point such that none of the components of $(D\times\mathbb R)\setminus\Gamma$ is extendable holomorphically to $a$. Then, $a$ is contained in an $n$-dimensional holomorphic graph lying on and closed in $\Gamma$. In particular, if $\Gamma$ divides two domains of holomorphy, then it is foliated by a family of closed holomorphic hypersurfaces–graphs. These results extend and generalize the well-known theorems of E. Levi, J.-M. Trépreau (proved for $C^2$-smooth $\Gamma$), and N. Shcherbina (proved for $n=1$). 
@article{TM_2001_235_a17,
     author = {E. M. Chirka},
     title = {Levi and {Tr\'epreau} {Theorems} for {Continuous} {Graphs}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {272--287},
     publisher = {mathdoc},
     volume = {235},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2001_235_a17/}
}
TY  - JOUR
AU  - E. M. Chirka
TI  - Levi and Tr\'epreau Theorems for Continuous Graphs
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2001
SP  - 272
EP  - 287
VL  - 235
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2001_235_a17/
LA  - ru
ID  - TM_2001_235_a17
ER  - 
%0 Journal Article
%A E. M. Chirka
%T Levi and Tr\'epreau Theorems for Continuous Graphs
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2001
%P 272-287
%V 235
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2001_235_a17/
%G ru
%F TM_2001_235_a17
E. M. Chirka. Levi and Tr\'epreau Theorems for Continuous Graphs. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 235 (2001), pp. 272-287. http://geodesic.mathdoc.fr/item/TM_2001_235_a17/

[1] Vladimirov V. S., Metody teorii funktsii mnogikh kompleksnykh peremennykh, Nauka, M., 1964 | MR

[2] Chirka E. M., “Priblizhenie golomorfnymi funktsiyami na gladkikh mnogoobraziyakh v $\mathbb{C}^n$”, Mat. sb., 78 (1969), 101–123 | Zbl

[3] Chirka E. M., “Priblizhenie mnogochlenami na zvezdnykh oblastyakh v $\mathbb{C}^n$”, Mat. zametki, 14 (1973), 55–60 | Zbl

[4] Chirka E. M., “Analiticheskoe predstavlenie $\mathrm{CR}$-funktsii”, Mat. sb., 98 (1975), 591–623 | Zbl

[5] Chirka E. M., Kompleksnye analiticheskie mnozhestva, Nauka, M., 1985 | MR

[6] Chirka E. M., “Vvedenie v geometriyu $\mathrm{CR}$-mnogoobrazii”, UMN, 46:1 (1991), 81–164 | MR | Zbl

[7] Chirka E. M., “Teorema Rado dlya $\mathrm{CR}$-otobrazhenii giperpoverkhnostei”, Mat. sb., 185:6 (1994), 125–144 | MR | Zbl

[8] Chirka E. M., “Golomorfnye sloeniya psevdovognutykh grafikov”, Dokl. RAN, 377:4 (2001), 452–454 | MR | Zbl

[9] Shabat B. V., Vvedenie v kompleksnyi analiz, Ch. 2, Nauka, M., 1985 | MR

[10] Hörmander L., Notions of convexity, Birkhäuser, Boston etc., 1994 | MR | Zbl

[11] Mañe R., Sad P., Sullivan D., “On the dynamics of rational maps”, Ann. Sci. Ecole Norm. Super, 16 (1983), 193–217 | MR | Zbl

[12] Shcherbina N. V., “On the polynomial hull of a graph”, Indiana Univ. Math. J., 42 (1993), 477–503 | DOI | MR | Zbl

[13] Trépreau J.-M., “Sur le prolongement holomorphe des fonctions C-R défines sur une hypersurface réelle de classe $C^2$ dans $\mathbb{C}^n$”, Invent. Math., 83 (1986), 583–592 | DOI | MR | Zbl