Germs of CR maps between real analytic hypersurfaces. (English) Zbl 0653.32020

Suppose H is a smooth CR mapping between two real-analytic hypersurfaces M and M’ in \({\mathbb{C}}^{n+1}\) and let \(P_ 0\in M\). The authors derive sufficient conditions for H to extend holomorphically to an ambient neighborhood of \(P_ 0\). The conditions are given in terms of algebraic concepts, essential finiteness of M at \(P_ 0\) and M’ at \(H(p_ 0)\), and finite multiplicity of H at \(P_ 0\). The techniques are rather algebraic as well, using such results as Nakayama’s Lemma, the Nullstellensatz, and the Weierstrass Preparation Theorem.
Reviewer: G.Harris


32V40 Real submanifolds in complex manifolds
Full Text: DOI EuDML


[1] D’Angelo, J.: The notion of formal essential finiteness for smooth real hypersurfaces. Indiana J. Math.36, 897-903 (1987) · Zbl 0628.32024
[2] Baouendi, M.S., Bell, S., Rothschild, L.P.: CR mappings of finite multiplicity and extension of proper holomorphic mappings. Bull. Am. Math. Soc.16, 265-270 (1987) · Zbl 0627.32016
[3] Baouendi, M.S., Bell, S., Rothschild, L.P.: Mappings of three-dimensional CR manifolds and their holomorphic extension. Duke Math. J. (to appear) · Zbl 0655.32015
[4] Baouendi, M.S., Jacobowitz, H., Treves, F.: On the analyticity of CR mappings. Ann. Math.122, 365-400 (1985) · Zbl 0583.32021
[5] Baouendi, M.S., Rothschild, L.P.: CR mappings and their holomorphic extension. Proceedings ?Journées Equations aux Derivées Partielles? Saint-Jean-de-Monts, June 1-5, 1987, Soc. Math. France
[6] Baouendi, M.S., Treves, F.: About the holomorphic extension of CR functions on real hypersurfaces in complex space. Duke Math. J.51, 77-107 (1984) · Zbl 0564.32011
[7] Bedford, E., Bell, S.: Extension of proper holomorphic mappings past the boundary. Manuscr. Math.50, 1-10 (1985) · Zbl 0583.32044
[8] Bell, S.: Analytic hypoellipticity of the \(\bar \partial \) problem and extendability of holomorphic mappings. Acta Math.147, 109-116 (1981) · Zbl 0475.32010
[9] Bell, S., Catlin, D.: Boundary regularity of proper holomorphic mappings. Duke Math. J.49, 385-396 (1982) · Zbl 0486.32014
[10] Bloom, T., Graham, I.: On ?type? conditions for generic submanifolds ofC?. Invent. Math.40, 217-243 (1977) · Zbl 0346.32013
[11] Derridj, M.: Le principe de réflexion en des points de faible pseudoconvexité pour des applications holomorphes propres. Invent. Math.79, 197-215 (1985) · Zbl 0554.32011
[12] Diederich, K., Fornaess, J.E.: Boundary regularity of proper holomorphic mappings. Invent. Math.67, 363-384 (1982) · Zbl 0501.32010
[13] Diederich, K., Webster, S.M.: A reflection principle for degenerate real hypersurfaces. Duke Math. J.47, 835-843 (1980) · Zbl 0451.32008
[14] Eisenbud, D., Levine, H.: An algebraic formula for the degree of aC ? map germ. Ann. Math.106, 19-44 (1977) · Zbl 0398.57020
[15] Fornaess, J.E.: Biholomorphic mappings between weakly pseudoconvex domains. Pac. J. Math.74, 63-65 (1978) · Zbl 0353.32026
[16] Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley-Interscience 1978 · Zbl 0408.14001
[17] Han, C.K.: Analyticity of C.R. equivalence between some real hypersurfaces inC n , with degenerate Levi form. Invent. Math.73, 51-69 (1983) · Zbl 0517.32007
[18] Kohn, J.J.: Boundary behaviour of \(\bar \partial \) on weakly pseudoconvex manifolds of dimension two. J. Differ. Geom.6, 523-542 (1972) · Zbl 0256.35060
[19] Lewy, H.: On the boundary behavior of holomorphic mappings. Acad. Naz. Lincei35, 1-8 (1977) · Zbl 0377.31008
[20] Nagata, M.: Local rings. New York: Wiley-Interscience 1962 · Zbl 0123.03402
[21] Pin?uk, S.I.: On proper holomorphic mappings of strictly pseudoconvex domains. Siberian Math. J.15, 909-917 (1974)
[22] Zariski, O., Samuel, P.: Commutative algebra, Vol. 1-2. New York: Van Nostrand 1958-1960 · Zbl 0081.26501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.