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A Morse-theoretical proof of the Hartogs extension theorem

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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 V(∂Ω) of a connected boundary ∂Ω ⋐ℂn ≥ 2) do extend holomorphically and uniquely to the domain ό. Martinelli, in the early 1940’s, and Ehrenpreis in 1961 obtained a rigorous proof, using a new multidimensional integral kernel or a short\(\bar \partial \) 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, Fornœss exhibited a topologically strange (nonpseudoconvex) domain όF ⊂ ℂ2 that cannot befitted in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside όF. 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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Authors and Affiliations

  1. Département de Mathématiques et Applications, UMR 8553 du CNRS, École Normale Supérieure, 45 rue d’Ulm, F-75230, Paris Cedex 05, France

    Joël Merker & Egmont Porten

  2. Department of Engineering, Physics and Mathematics, Mid Sweden University, Campus Sundsvall, S-85170, Sundsvall, Sweden

    Joël Merker & Egmont Porten

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  1. Joël Merker
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  2. Egmont Porten
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Correspondence to Joël Merker.

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Communicated by Steven Krantz

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Merker, J., Porten, E. A Morse-theoretical proof of the Hartogs extension theorem. J Geom Anal 17, 513–546 (2007). https://doi.org/10.1007/BF02922095

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