Prolongements analytiques de la solution du problème de Cauchy linéaire (Q1068257)

[Shortened version of the original review.] Let \(\Omega\) be a connected, paracompact, not compact, complex analytic manifold; let \(\omega\) be any of its points. Let \(\Sigma\) be a simply connected, paracompact, not compact Riemann surface; let \(\sigma\) be any of its points. Let \(\alpha\) be some given point of \(\Sigma\) ; denote: \(x = (\sigma,\omega)\in X = \Sigma \times \Omega\); \(\alpha \times \Omega = \{(\sigma,\omega)\in \Sigma \times \Omega\); \(\sigma = \alpha \}.\) Let A be a differential operator of order m, holomorphic in some neighborhood of \(\alpha\times \Omega\), such that no hypersurface \(\sigma\times \Omega\) is characteristic at some of its points. By the datum of two numerical functions v and w, holomorphic in some neighbourhood of \(\alpha\times \Omega\), define the Cauchy problem, whose unknown is u: (1) \(Au=v,\) \(u-w\) vanishes m-times on \(\alpha\times \Omega;\) (or the similar one, where u, v and w are vector-valued and A is a matrix). The aim of the paper is as simply as possible constructions of as large as possible domains of X, on which the problem (1) has a holomorphic solution u. For that purpose the author constructs a Riemannian metric \(ds^ 2\) and defines dist(\(\sigma,\alpha)=\min \int^{\sigma}_{\alpha}ds\) on \(\Sigma\). That dist appears in the three main results. Some tools of the proofs are an appropriate version of Cauchy-Kowalewski theorem and analytic continuation using germs and based on sheaf theory.