2004/04/13 by Joël Merker, Merker, Joël
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #math.CV #math.DG
paper · pdf · doi:10.48550/arxiv.math/0404248
arxiv created 2004/04/13 · openalex publication_date 2004/04/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Searching normal forms for real analytic submanifolds of Cn involves convergence problems. In 1983, J.K. Moser and S.M. Webster provided examples of real analytic surfaces in C2 having an isolated hyperbolic (in the sense of E. Bishop) complex tangency, which are formally but not holomorphically normalizable (because of the presence of small divisors), even if the normal form is itself real analytic or algebraic. On the contrary, it appears that such a nonconvergence phenomenon does not appear for submanifolds of Cn whose CR dimension is locally constant, in view of recent results by S.M. Baouendi, P. Ebenfelt and L.P. Rothschild. These results hold true with hypotheses which are relatively simple, but satisfied at a Zariski-generic. Notably, these authors establish that every invertible formal CR mapping between two submanifolds of Cn which are real analytic, generic, finitely nondegenerate and minimal (in the sense of J.-M. Trépreau and A.E. Tumanov) is convergent. In this paper, we establish a more general convergence theorem, which is valid without any nondegeneracy condition, and which confirms the rigidity of the CR category (see Theorem 1.23). This result may be interpreted as a formal Schwarz reflection principle for CR mappings. We deduce that every formal CR equivalence between two submanifolds of Cn which are real analytic, generic and minimal is convergent if and only if both submanifolds are holomorphically nondegenerate (in the sense of N. Stanton). Finally, we establish that two submanifolds of Cn which are real analytic, generic and minimal are formally CR equivalent if and only if they are biholomorphically equivalent.