Probl\`eme de Plateau complexe feuillet\'e. Ph\'enom\`enes de Hartogs-Severi et Bochner pour des feuilletages CR singuliers

TL;DR

This paper extends classical theorems on analytic continuation to complex foliated structures, demonstrating how real analytic functions harmonic on Levi-flat annuli can be extended within complex analytic sets.

Contribution

It generalizes Severi, Brown, and Bochner's theorems to singular CR foliations, providing new extension results for harmonic functions on Levi-flat structures.

Findings

Extension of harmonic functions on Levi-flat annuli to complex analytic sets.

Existence of Levi-flat real analytic subsets filling given Levi-flat annuli.

Extension results for functions with prescribed boundary cycles.

Abstract

The purpose of this paper is to generalize in a geometric setting theorems of Severi, Brown and Bochner about analytic continuation of real analytic functions which are holomorphic or harmonic with respect to one of its variables. We prove in particular that if N is a real analytic levi-flat annulus in an open set of R^{n}\timesC^{2}, then one can find X\subsetR^{n}\timesC^{2} such that X\cupN is a levi-flat real analytic subset and X fills N in the sense that the boundary of the integration current of X is a prescribed smooth submanifold of N foliated by real curves. Moreover, real analytic functions on N whose restrictions to complex leaves are harmonic extend to X in functions of the same kind. We give also a theorem when the prescribed boundary is a cycle.