An extension of the theorem of Hartogs

Abstract:

Hartogs proved that every function which is holomorphic on the boundary of the unit ball in ${C^n},n > 1$, can be extended to a function holomorphic on the ball itself. It is conjectured that a real k-dimensional ${\mathcal {C}^\infty }$ compact submanifold of ${C^n},k > n$, is extendible over a manifold of real dimension $(k + 1)$. This is known for hypersurfaces (i.e., $k = 2n - 1$) 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.