On Levi-flat hypersurfaces with given boundary in ℂ n

Abstract

Let S ⊂ C n be a compact connected 2-codimensional submanifold. If n 3, essentially local conditions and the assumption: every complex point of S is elliptic imply the existence of a projection in C n of a Levi-flat (2n − 1)-subvariety whose boundary is S (Dolbeault, Tomassini, Zaitsev, 2005). We extend the result when S is homeomorphic to a sphere and has one hyperbolic point.For n = 2 many results are known since the 1980's and a new result with a very technical hypothesis is announced.

CR Singularities and Generalizations of Moser's Theorem I

“…The C.-R. Singularities [4], [6], [11], [14], [15], [17], [26] in codimension 2 are important for the area of the analysis of several complex variables. Dolbeault [8], [9], Dolbeault-Tomassini-Zaitsev [10], [11] used the existence of the C.-R. Singularities in order to study the problems of existence and uniqueness of Levi-flat hypersurfaces with prescribed compact boundary [10], [11]. The author [6] constructed a family of analytic discs attached to a class of C.-R. Singular real submanifolds in codimension 2 trying to understand the local hull of holomorphy using methods from Huang-Krantz [15].…”

Boundaries of Levi-flat hypersurfaces: special hyperbolic points

“…Proposition 14. [cf [Dol08][Proposition 2.6.1]] Let S ⊂ C n be a compact connected real 2-codimensional manifold such that the following holds:…”

“…Then we consider elementary models and their gluing to obtain more complicated examples (section 3.5). Results have been announced in [Dol08], and in more precise way in [Dol11]; the first aim of this paper is to give complete proofs. Finally, we recall in detail and extend the results of [DTZ09] on regularity of the solution when S is a graph satisfying a supplementary global condition, as in the case n = 2, to the case of existence of special 1-hyperbolic points, and to gluing of elemetary smooth models (section 4).…”

“…Then we consider elementary models and their gluing to obtain more complicated examples (section 3.5). Results have been announced in [Dol08], and Date: November 13, 2018.…”