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A characterization of totally real generic submanifolds of strictly pseudoconvex boundaries in \({\mathbb{C}}^ n\) admitting a local foliation by interpolation submanifolds. (English) Zbl 0716.32013
Let D be a domain in \({\mathbb{C}}^ n\). A submanifold \(M\subset \partial D\) is called an interpolation submanifold if for every \(p\in M\) the tangent space \(T_ p(M)\) is contained in the maximal complex subspace of \(T_ p(\partial D)\). The existence of foliations of \(\partial D\) by interpolation submanifolds is important in characterization of peak and maximum modulus sets on \(\partial D\). The author proves that each 2- dimensional totally real submanifold of \(\partial D\) admits local foliations by interpolation submanifolds. Here D is a domain in \({\mathbb{C}}^ 2\) with \(C^ 1\) boundary. For \({\mathbb{C}}^ n\), \(n\geq 3\), a characterization of totally real sets admitting such foliations is given in terms of the Levi form. Applications to description of maximum modulus sets on \(\partial D\) for strictly pseudoconvex domais are also given.
Reviewer: A.Russakovskij

MSC:
32V40 Real submanifolds in complex manifolds
32T99 Pseudoconvex domains
32A38 Algebras of holomorphic functions of several complex variables
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References:
[1] Burns, D., Stout, E.L.: Extending functions from submanifolds of the boundary. Duke Math. J.43, 391-404 (1976) · Zbl 0328.32013 · doi:10.1215/S0012-7094-76-04335-0
[2] Chaumat, J., Chollet, A.M.: Charactérisation et propriétés des ensembles localement pics deA ? (D). Duke Math. J.47, 763-787 (1980) · Zbl 0454.32013 · doi:10.1215/S0012-7094-80-04745-6
[3] Duchamp, Th., Stout, E.L.: Maximum modulus sets. Ann. Inst. Fourier47, 37-69 (1981) · Zbl 0439.32007
[4] Fornaess, J.E., Henriksen, B.S.: Characterization of global peak sets forA ?(D). Math. Ann.259, 125-130 (1982) · Zbl 0489.32010 · doi:10.1007/BF01456835
[5] Hakim, M., Sibony, N.: Ensembles pics dans des domaines strictement pseudoconvexes. Duke Math. J.45, 601-617 (1978) · Zbl 0402.32008 · doi:10.1215/S0012-7094-78-04527-1
[6] Iordan, A.: Ensembles de module maximal dans des domaines pseudoconvexes. C. R. Acad. Sc. Paris 300, Sér. I,19, 655-656 (1985) · Zbl 0585.32021
[7] Stout, E.L.: Interpolation manifolds. Recent developments in several complex variables. Ann. Math. Stud. (1981)
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