Some remarks concerning holomorphically convex hulls and envelopes of holomorphy. (English) Zbl 0816.32011

Let \(\Omega\) be a bounded strictly convex domain in \(\mathbb{C}^ 2\) with smooth boundary, and let \(K\) be a compact subset of the boundary \(\partial \Omega\). Stout, Lupacciolu and Alexander proved that each continuous \(CR\)-function on \(\partial \Omega \backslash K\) has uniquely determined analytic extension to \(\Omega \backslash \widehat K\), \(\widehat K\) being the polynomial hull of \(K\).
There is a heuristic way to see this, namely by observing the connection between Oka’s characterization principle for polynomial hulls on the one hand and analytic continuation along continuous one-parameter families of one-dimensional analytic varieties on the other hand. Monodromy considerations for proving that the envelope of holomorphy is schlicht are based on the fact that in \(\mathbb{C}^ 2\) the complex codimension of the considered varieties is equal to one. These observations are managed into a rigorous proof, which differs from the original one.
The result is generalized to arbitrary domains of holomorphy with \(C^ 2\) boundary, the polynomial hull in that case to be replaced by the hull with respect to the space of functions which are holomorphic in \(\Omega\) and continuous in the closure \(\overline \Omega\). Previous results of this kind concern the case of domains \(\Omega\) with \(\overline \Omega\) having a Stein neighbourhood basis and the (in general larger) hull with respect to functions which are analytic in a neighbourhood of \(\overline \Omega\).


32D10 Envelopes of holomorphy
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
32D20 Removable singularities in several complex variables
32V05 CR structures, CR operators, and generalizations
32E35 Global boundary behavior of holomorphic functions of several complex variables
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