On the analyticity of smooth CR mappings between real-analytic CR manifolds
The main theorem of this article provides a synthesis of a great number of results about the analyticity of smooth CR mappings, starting with the classical Lewy-Pinchuk principle. It also provides an entirely new proof of the Baouendi-Jacobowitz-Trèves analyticity theorem [\textit{M. S. Baouendi, H. Jacobowitz} and \textit{F. Trèves}, Ann. Math. (2) 122, 365-400 (1985; Zbl 0583.32021)], without any rank condition on the mapping. As a matter of fact, essentially all smooth reflection principles obtained by Baouendi-Rothschild and others are covered by the main theorem of the article under review. The general framework of this work was introduced in two previous articles by \textit{B. Coupet, S. Pinchuk} and \textit{A. Sukhov} [C. R. Acad. Sci., Paris Sér. I, Math. 329, No. 6, 489-494 (1999; Zbl 0949.32018) and Math. Z. 235, No. 3, 541-557 (2000; Zbl 0972.32008)] where the problem is dealt with for a codimension one source CR manifold. Let \(f: M\to M'\) be a \({\mathcal C}^\infty\)-smooth CR mapping between two real analytic generic submanifolds of \({\mathbb C}^n\) and \({\mathbb C}^{n'}\). Let \(p\in M\) and assume that \(M\) is minimal in the sense of Trépreau-Tumanov at \(p\), so CR functions extend holomorphically to a wedge at \(p\). Let \(\overline{L}_1,\dots,\overline{L}_m\) be a basis of \((0,1)\) vector fields tangent to \(M\), where \(m=\text{ CRdim} M\). Define the first characteristic variety associated with \(f\) to be \[ {\mathbf V}_p':=\{z'\in {\mathbb C}^{n'}:\overline L^\alpha \rho_k'(z', \overline{f(\cdot)}) |_p=0, \forall \alpha\in {\mathbb N}^m, \forall k=1, \dots,d'\}, \] where \(\rho_k'(z',\bar z')=0\), \(k=1,\dots,d'\) are defining equations for \(M'\) and where \(p'=f(p)\). We have \(p'\in {\mathbf V}_p'\). The main result states that if the dimension of \({\mathbf V}_p'\) is zero at \(p'\), then \(f\) extends holomorphically to a neighborhood of \(p\). Two main ideas are used in the proof. The first one is to deal abstractly with some fields generated by the components \(f_1,\dots, f_{n'}\) of the mapping \(f\) and by the jets of its conjugate components and to come down to a result about meromorphic extension of a quotient of two such functions. The second one is to introduce some wedgelike domains attached to \(M\) in order to bypass the subtle point that in Tumanov's extension theorem, the directions of the wedges of extension are not controlled and do depend on the neighborhood to which analytic discs which fill them are attached. After this article was published, \textit{N. Mir, F. Meylan} and \textit{D. Zaitsev} [Math. Res. Lett. 9. No. 1, 73-93 (2002; Zbl 1009.32024)] proved a similar result following the same interesting strategy. Further developments of the article under review are provided in a recently published article [\textit{B. Coupet, S. Damour, J. Merker} and \textit{A. Sukhov}, C. R. Acad. Sci., Paris Sér. I, Math. 334, No. 11, 953 --956 (2002; Zbl 1010.32019)].
minimality in the sense of Trépreau-Tumanov
scientific article; zbMATH DE number 1742562