An extension of the theorem of Hartogs

Abstract

Hartogs proved that every function which is holomorphic on the boundary of the unit ball in C n , n > 1 {C^n},n > 1 , can be extended to a function holomorphic on the ball itself. It is conjectured that a real k-dimensional C ∞ {\mathcal {C}^\infty } compact submanifold of C n , k > n {C^n},k > n , is extendible over a manifold of real dimension ( k + 1 ) (k + 1) . This is known for hypersurfaces (i.e., k = 2 n − 1 k = 2n - 1 ) and submanifolds of real codimension 2. It is the purpose of this paper to prove this conjecture and to show that we actually get C-R extendibility.