Boundaries of singularity sets, removable singularities, and CR-invariant subsets of CR-manifolds

Cr-Meromorphic Extension and the Nonembeddability of the Andreotti–rossi Cr Structure in the Projective Space

“…This problem is related to the study of the decomposition of M into CR orbits. Indeed, if M is composed of a single CR orbit, by propagation of CR extension, CR functions on M extend to a one-sided neighborhood of M. Deforming M into this one sided neighborhood, we can assume that CR functions on M are restriction of holomorphic functions in a connected neighborhood of M. Thus, by Theorem 3, we have extension to one side of M. According to[20], if M c C, this is automatically verified. More, suppose that there exists a complex hypersurface H C P n {C) such that the Levi form of M does not vanish on MflH.…”

“…Then we prove that the indeterminacy set is metrically thin so that all the theory of CR extension and CR removability [26,17,5,20,27,28,8] apply. We obtain another proof of our extension theorem and a generalization of the notion of CR-meromorphic maps to generic CR submanifolds of C n .…”

“…By the Veronese embedding, P n (C)\H can be embedded in C m (for some m). Then, the embedding of M\H being a maximally complex submanifold of C m with compact Levi flat part, M is composed of a single CR orbit (see[20]). Finally, the general result would be obtained if the answer to the following question is positive: Let M be a smooth real hypersurface of P n (C) (n > 2) making a partition of P n (C) into two open sets.…”

Totally real discs in non-pseudoconvex boundaries

“…a chain of smooth curves tangent to T~3I. Bv a fundamental observation of H. Sussmann [28], [14], a CR orbit is either an open subset of 21I or an injectively immersed Riemann surface, and the union of all CR orbits of codimension one is relatively closed in M. If 21.I has only one CR orbit, it is called globally minimal. We recall from [14] the fact which we use in the reduction of Theorem 1 to Theorem 2.…”

“…Let us briefly give some indications about formal generalizations for n_>3. If one replaces S with a totally real ball of maximal real dimension n, removability becomes a corollary of more general results: For n=3, it is a consequence of a theorem of JSricke [14] on real submanifolds of codimension two in the boundary. For dimension r~_>4, it follows from a result of Lupacciolu and Stout [22] about the removability of metrically thin singularities.…”