A Morse-theoretical Proof of the Hartogs Extension Theorem
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چکیده
100 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 b C (n > 2) do extend holomorphically and uniquely to the domain Ω. 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, Fornæss in 1998 exhibited a topologically strange (nonpseudoconvex) domain Ω ⊂ C that cannot be filled in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside Ω. However, one should point out that the standard, unrestricted disc method usually allows discs to go outsise the domain (just think of Levi pseudoconcavity). Using the method of analytic discs for local extensional steps and Morsetheoretical tools for the global topological control of monodromy, we show that the Hartogs extension theorem can be established in such a way. Table of contents 1. The Hartogs extension theorem and the method of analytic discs . . . . . . . . . . . . 1. 2. Preparation of the boundary and unique extension . . . . . . . . . . . . . . . . . . . . . . . . . 5. 3. Quantitative Hartogs-Levi extension by pushing analytic discs . . . . . . . . . . . . . . 9. 4. Filling domains outside balls of decreasing radius . . . . . . . . . . . . . . . . . . . . . . . . . 13. 5. Creating domains, merging and suppressing connected components . . . . . . . . 20. 6. The exceptional case kλ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30. [22 colored illustrations] §1. THE HARTOGS EXTENSION THEOREM AND THE METHOD OF ANALYTIC DISCS 100 years ago exactly, in 1906, the publication of Hartogs’s thesis ([14] under the direction of Hurwitz) revealed what is now considered to be the most striking fact of multidimensional complex analysis: the automatic, compulsory holomorphic extension of functions of several complex variables to larger domains, especially for a class of “pot-looking” domains, nowadays called Hartogs figures, that may be filled in up to their top. Soon after, E.E. Levi [25] applied the Hurwitz-Hartogs argument of Cauchy integration on complex affine
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