Journal article

A morse-theoretical proof of the Hartogs extension theorem

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Author list: Porten, Egmont

Publication year: 2007

Start page: 513

End page: 546

Number of pages: 34

ISSN: 1050-6926

View additional information: View in Web of Science™

Abstract

One hundered years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was understood later: Holomorphic functions in a connected neighborhood nu(partial derivative Omega) of a connected boundary a partial derivative Omega C subset of C-n (n >= 2) do extend holomorphically and uniquely to the domain Omega. Martinelli, in the early 1940s, and Ehrenpreis in 1961 obtained a rigorous proof using a new multidimensional integral kernel or a short argument, but it remained unclear how to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906), and E E. Levi (1911) in some special, model cases. In fact, known attempts (e.g., Osgood, 1929, Brown, 1936) struggled for monodromy against multivaluations, but failed to get the general global theorem.

Moreover, quite unexpectedly, in 1998, Fornaess exhibited a topologically strange (nonpseudoconvex) domain Omega(F) subset of C-2 that cannot befilled in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside Omega F. However one should point out that the standard, unrestricted disc method usually allows discs to go outside the domain (just think of Levi pseudoconcavity).

Using the method of analytic discs for local extensional steps and Morse-theoretical tools for the global topological control of monodromy, we show that the Hartogs extension theorem can be established in such a way.

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