Extension problems in complex and CR-geometry.

*(English)*Zbl 1165.32001
Tesi. Scuola Normale Superiore Pisa (Nuova Serie) 9. Pisa: Edizioni della Normale; Pisa: Scuola Normale Superiore (Diss. 2007) (ISBN 978-88-7642-338-3/pbk). xiv, 153 p. (2008).

The volume contains the Ph.D. thesis of the author. The exposition is divided into three parts. The first one contains a review of the basis notions and the classical extension results in complex and CR geometry, starting from the results of Hans Lewy on boundary values of holomorphic functions of two complex variables. The second and third part contain the original results, also obtained in collaboration with G. Tomassini and G. Della Sala [Math. Z. 256, No. 4, 737–748 (2007; Zbl 1118.32010); Bull. Sci. Math. 132, No. 3, 232–245 (2008; Zbl 1144.32015); Int. J. Math. 18, No. 2, 203–218 (2007; Zbl 1140.32025)]. A main topic are the \(q\)-coronae, and also slightly more general domains, for which various results are proved, concerning holomorphic and meromorphic functions and divisors. Next, he turns to the problem of finding conditions for an odd dimensional real submanifold of a complex manifold to be the boundary of an embedded complex submanifold. After reviewing the already known results for the compact case, the author discusses his most recent contributions to the study of the non-compact case. The last chapter contains results in collaboration with G. Della Sala that are not yet published. The question, to which they give some partial answer, is that of finding, given a pseudoconvex domain \(\Omega\) of \(\mathbb{C}^n\) and an open subset \(A\) of its boundary \(\partial\Omega\), an optimal domain \(D\subset\Omega\), to which all odd-dimensional maximally complex submanifolds \(M\) of \(A\) can be extended.

Reviewer: Mauro Nacinovich (Roma)