Abstract
It is proved that CR functions on a quadratic cone M in \({\mathbb{C}^n}\), n > 1, admit one-sided holomorphic extension if and only if M does not have two-sided support, a geometric condition on M which generalizes minimality in the sense of Tumanov. A biholomorphic classification of quadratic cones in \({\mathbb{C}^2}\) is also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andreotti, A., Denson Hill, C., Levi, E.E.: Convexity and the Hans Lewy problem. I. Reduction to vanishing theorems. Ann. Scuola Norm. Sup. Pisa 26(3), 325–363 (1972)
Baouendi, M.S., Rothschild, L.P.: Cauchy–Riemann functions on manifolds of higher codimension in complex space. Invent. Math. 101(1), 45–56 (1990)
Baouendi, M.S., Trèves, F.: A property of the functions and distributions annihilated by a locally integrable system of complex vector fields. Ann. Math. 113(2), 387–421 (1981)
Bedford, E., Fornæss, J.E.: Local extension of CR functions from weakly pseudoconvex boundaries. Michigan Math. J. 25(3), 259–262 (1978)
Burns, D., Gong, X.: Singular Levi-flat real analytic hypersurfaces. Am. J. Math. 121(1), 23–53 (1999)
Cirka, E.M.: Analytic representation of CR-functions. Mat. Sb. (N.S.) 98(140), (1975), no. 4(12), 591–623, 640 [English translation in Math. USSR Sbornik 27(4), 526–553 (1975)]
Chirka, E.M.: Complex analytic sets. Mathematics and its Applications (Soviet Series), vol. 46. Kluwer, Dordrecht (1989)
Chirka, E.M.: Levi and Trépreau theorems for continuous graphs. Tr. Mat. Inst. Steklova 235 (2001), Anal. i Geom. Vopr. Kompleks. Analiza, 272–287 [translation In: Proc. Steklov Inst. Math. 235(4), 261–276 (2001)]
Chirka, E.M., Stout, E.L.: Removable singularities in the boundary. Contributions to complex analysis and analytic geometry. Aspects Math. E26, Vieweg, Braunschweig, pp. 43–104 (1994)
Chirka, E.M.: Rado’s theorem for CR-mappings of hypersurfaces. Mat. Sb. 185(6), 125–144 (1994) [English translation in Russian Acad. Sci. Sb. Math. 82(1), 243–259 (1995)]
Ermolaev, Ju.B.: The simultaneous reduction of symmetric and Hermitian forms. Izv. Vysš. Učebn. Zaved. Matematika 21(2), 10–23 (1961)
Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Band 153 Springer-Verlag New York Inc., New York (1969)
Kohn, J.J., Nirenberg, L.: A pseudo-convex domain not admitting a holomorphic support function. Math. Ann. 201, 265–268 (1973)
Kytmanov, A., Myslivets, S., Tarkhanov, N.: Removable singularities of CR functions on singular boundaries. Math. Z. 242(3), 491–515 (2002)
Merker, J., Porten, E.: Holomorphic extension of CR functions, envelopes of holomorphy, and removable singularities. Int. Math. Res. Surv. 1–287 (2006)
Rosay, J.-P.: À propos de “wedges” et d’“edges”, et de prolongements holomorphes. Trans. Am. Math. Soc. 297(1), 63–72 (1986)
Shcherbina, N.V.: On the polynomial hull of a graph. Indiana Univ. Math. J. 42(2), 477–503 (1993)
Takagi, T.: On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau. Japan J. Math. 1, 83–93 (1924) [Also reprinted in his Collected Works. Springer, Tokyo, pp. 226-234 (1990)]
Trépreau, J.-M.: Sur le prolongement holomorphe des fonctions C-R défines sur une hypersurface réelle de classe C 2 dans C n. Invent. Math. 83(3), 583–592 (1986)
Tumanov, A.E.: Extension of CR-functions into a wedge from a manifold of finite type. Mat. Sb. (N.S.) 136(178)(1), 128–139 (1988) [translation in Math. USSR-Sb. 64(1), 129–140 (1989)]
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article is available at http://dx.doi.org/10.1007/s00208-009-0369-x.
Rights and permissions
About this article
Cite this article
Chakrabarti, D., Shafikov, R. Holomorphic extension of CR functions from quadratic cones. Math. Ann. 341, 543–573 (2008). https://doi.org/10.1007/s00208-007-0200-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-007-0200-5