A geometrical proof of the Hartogs extension theorem

Abstract

100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was stated in full generality later by Osgood (1929): holomorphic functions in a connected neighborhood V(bD) of a connected boundary bD contained in C^n (n >= 2) do extend holomorphically and uniquely to the domain D. It was a long-standing open problem to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906) and E.E. Levi (1911) in some special, model cases. Quite unexpectedly, Fornaess in 1998 exhibited a topologically strange (nonpseudoconvex) domain D^F in C^2 that cannot be filled in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside D^F. However, one should point out that the standard, unrestricted disc method usually allows discs to go outsise the domain (just think of Levi pseudoconcavity). Confirming these rather ancient expectations, we show that the global Hartogs extension theorem can be established in such a natural way.