Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

Abstract

We prove that generic homologically nontrivial $(2n-1)$-parameter family of analytic discs attached by their boundaries to a CR manifold $\Omega$ in $\mathbb C^n, n \le 2$ tests CR functions: if a smooth function on $\Omega$ extends analytically inside each analytic disc then it satisfies the tangential CR equations. In particular, we answer, in real analytic category, two open questions: on characterization of analytic functions in planar domains (the strip-problem), and on characterization of boundary values of holomorphic functions in domains in $\mathbb C^n$ (a conjecture of Globevnik and Stout). We also characterize complex curves in $\mathbb C^2$ as real 2-manifolds admitiing homologically nontrivial 1-parameter families of attached analytic discs. The proofs are based on reduction to a problem of propagation of degeneracy of CR foliations of torus-like manifolds.