Characteristic foliations on maximally real submanifolds of Cⁿ and envelopes of holomorphy

الأصل · EN

Let S be an arbitrary real surface, with or without boundary, contained in a hypersurface M of the complex euclidean space ², with S and M of class C², ᵃ, where 0 < a < 1. If M is globally minimal, if S is totally real except at finitely many complex tangencies which are hyperbolic in the sense of E. Bishop and if the union of separatrices is a tree of curves without cycles, we show that every compact K of S is CR-, W- and Lᵖ-removable (Theorem 1.3). We treat this seemingly global problem by means of purely local techniques, namely by means of families of small analytic discs partially attached to maximally real submanifolds of Cⁿ and by means of a thorough study of the relative disposition of the characteristic foliation with respect to the track on M of a certain half-wedge attached to M. This localization procedure enables us to answer an open problem raised by B. Jöricke: under a certain nontransversality condition with respect to the characteristic foliation, we show that every closed subset C of a C²,ᵃ-smooth maximally real submanifold M¹ of a (n-1)-codimensional generic C²,ᵃ-smooth submanifold of ⁿ is CR-, W- and Lᵖ-removable (Theorem 1.2'). The known removability results in CR dimension at least two appear to be logical consequences of Theorem 1.2'. The main proof (65p.) is written directly in arbitrary codimension. Finally, we produce an example of a nonremovable 2-torus contained in a maximally real 3-dimensional maximally real submanifold, showing that the nontransversality condition is optimal for universal removability. Numerous figures are included to help readers who are not insiders of higher codimensional geometry.