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International Mathematics Research Papers (2006) Vol. 2006 : article ID 72069, 131 pages, doi:10.1155/IMRP/2006/72069
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Characteristic foliations on maximally real submanifolds of Cn and removable singularities for CR functions

Joël Merker and Egmont Porten

Département de Mathématiques et Applications UMR 8553 du CNRS, École Normale Supérieure, 45 rue de Ulm, 75230 Paris Cedex 05, France E-mail address: merker{at}dma.ens.fr
Department of Engineering, Physics and Mathematics, Mid Sweden University Campus Sundsvall, 85170 Sundsvall, Sweden E-mail address: egmont.porten{at}miun.se

In present-day multidimensional complex analysis, the existing cohomological or functional characterizations of removable singularities (for holomorphic or CR functions) do only seldom provide adequate insights into the geometrical structures. Nonetheless, in the theory of CR functions, some geometric criteria are accessible for Lp-removability in the spirit of the classical Denjoy theorem (cf. the Painlevé problem), especially in the case of CR dimension 1, where, as in the complex plane, a single Formulab operator is concerned. We consider closed or compact singularities a priori, contained in some surface S embedded into a globally minimal hypersurface M sub C2 (geometric assumptions). If S is totally real except at finitely many complex tangencies that are hyperbolic in the sense of Bishop, and if the union of the separatrices of its characteristic foliation is a tree of curves having no cycles, we show that every compact set K sub S is removable. Already in the hypersurface case, we endeavor a new localization procedure yielding substantial generalizations of this statement, for the removability of closed sets C sub M1 sub M contained in a totally real 1-codimensional submanifold M1 embedded in some C2,{alpha} (0 < {alpha} < 1) generic submanifold M sub Cn(n >= 2) that has CR dimension 1. We establish that every characteristically pseudoconcave subset C sub M1 sub M closed both in M1 and in M is removable.


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