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A morse-theoretical proof of the Hartogs extension theorem
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
2007 (English)In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 17, no 3, 513-546 p.Article in journal (Refereed) Published
Abstract [en]

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 1940's, 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.

Place, publisher, year, edition, pages
2007. Vol. 17, no 3, 513-546 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-8433ISI: 000250826000008Scopus ID: 2-s2.0-84867978570OAI: oai:DiVA.org:miun-8433DiVA: diva2:139626
Available from: 2009-01-25 Created: 2009-01-25 Last updated: 2017-12-14Bibliographically approved

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Porten, Egmont

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