Maresin, I. V.; Sergeev, A. G. A microlocal version of Cartan-Grauert’s theorem. (English) Zbl 0937.32016 Ann. Pol. Math. 70, 157-162 (1998). Cartan proved the following theorem: Theorem. Let \(\omega\) be a domain in \({\mathbb R}^n= {\mathbb R}^n+i0\subset{\mathbb C}^n\) then \(\omega\) has a fundamental system of neighborhoods in \({\mathbb C}^n\) which are domains of holomorphy. Building on earlier work of Bros and Iagolnitzer, the authors extend this result to the case of tuboids. These are domains in \({\mathbb C}^n\) with a totally real edge which look asymptotically, near the edge, like a tube over a convex cone. A manifold \(M\subset{\mathbb C}^n\) is strictly real like if it is the graph of a smooth map \(F:{\mathbb R}_{(x)}^n\to {\mathbb R}^n_{(y)}\) with \(\|F'(x)\|<1\) for all \(x\in{\mathbb R}^n_{(x)}.\) The authors state: Theorem. Let \(M\) be a \(C^2\)-smooth strictly real-like submanifold in \({\mathbb C}^n\) and \(\omega\) a domain on \(M.\) Suppose that \(\Lambda\) is a fiberwise convex profile over \(\omega.\) For any tuboid \(\Omega'\) with profile \(\Lambda\) there exists a tuboid \(\Omega\subset\Omega'\) with the same profile which is a domain of holomorphy in \({\mathbb C}^n.\) An outline of the proof is given. Reviewer: Ch.Epstein (Philadelphia) MSC: 32V40 Real submanifolds in complex manifolds 32D05 Domains of holomorphy Keywords:totally real submanifolds; domains of holomorphy; tuboids PDFBibTeX XMLCite \textit{I. V. Maresin} and \textit{A. G. Sergeev}, Ann. Pol. Math. 70, 157--162 (1998; Zbl 0937.32016) Full Text: DOI