Projective geometry and Riemann's mapping problem
We quote from the introduction: ``A main purpose of this paper is to point out the rôle that projective geometry plays in complex function theory. \dots Segre suggested the consideration of the complex hypersurfaces \(r(z^\alpha, w^\alpha) = 0\), regarding the \(w^\sigma\) as parameters. With these hypersurfaces as a generalization of hyperplanes Hachtroudi [\textit{M. Hachtroudi}: Les espaces d'éléments a connexion projective normale, Hermann, Paris (1937; Zbl 0019.30601)] in 1937 introduced a projective connection or a generalized projective geometry. Such a projective structure will be within the holomorphic category and should be very useful, but we do not know how to define one for general domains. Instead we will restrict ourselves to a particular problem, viz., the problem of a generalization of Riemann's mapping theorem to higher dimensions. \dots In 1975, the first author showed [Math. Scand. 36, 74-82 (1975; Zbl 0305.53019)] that a projective connection, called Hachtroudi connection in this paper, can be intrinsically defined over the Segre family associated to a real-analytic hypersurface in \(\mathbb{C}^{n + 1}\), which is a generalization of an early work of Hachtroudi [loc. cit.] and of the classical projective geometry. In the \(CR\)-manifold theory, a domain \(D\) in \(\mathbb{C}^{n + 1}\) whose Cartan connection is flat over \(\partial D\) may not be globally biholomorphic t o \(B^{n + 1}\), by the example of Burns-Shnider [\textit{D. Burns} and \textit{S. Shnider}, Proc. Symp. Pure Math. 30, Part 2, Williamstown 1975, 141-168 (1977; Zbl 0422.32016)]. However, for a Segre family \({\mathcal M}_D\), when its Hachtroudi connection is flat, we obtain the following global result. Theorem 1. Let \(D \subset \mathbb{C}^{n + 1}\) be a bounded domain whose boundary \[ \partial D := \{(z^\alpha, w) \in \mathbb{C}^n \times \mathbb{C}\mid r(z^\alpha, w, \overline{z}^\alpha, \overline {w})= 0\} \] is a smooth connected real-analytic hypersurface, where \(r\) is a real-analytic function defined on \(\mathbb{C}^{n + 1}\). Then the following three statements are equivalent: (i) There is a holomorphic mapping \(g : \mathbb{C}^{n + 1} \to \mathbb{C}^{n + 1}\) such that the Jacobian \(\text{det}(Dg) \equiv \) constant and that the restriction \(g : D \to B^{n + 1}\) is biholomorphic. (ii) The Segre family \({\mathcal M}_D\) is defined on \(\mathbb{C}^{2n + 2}\), smooth, liftable, nondegenerate and locally isomorphic to the Segre family \({\mathcal M}_{B^{n + 1}}\), and there is one biholomorphic mapping \(g : U_{(0)} \to \widetilde{U}_{(0)}\) such that \(g(\partial D \cap U_{(0)}) \subset \partial B^{n + 1} \cap \widetilde{U}_{(0)}\) and such that \(\text{det} (Dg) \equiv \) constant, where \(U_{(0)}\) are open subsets in \(\mathbb{C}^{n + 1}\) with \(\partial D \cap U_{(0)} \neq \emptyset\) and \(\partial B^{n + 1} \cap \widetilde{U}_{(0)} \neq \emptyset\). (iii) \({\mathcal M}_D\) is defined on \(\mathbb{C}^{2n + 2}\), smooth, liftable, non-degenerate, and the associated Hachtroudi connection \(\pi_D\) is flat, and there is one local biholomorphic mapping \(g\) as in (ii). Moreover, from (ii) or (iii), the local mapping \(g\) can be extended into the global mapping \(g\) in (i).
scientific article; zbMATH DE number 799561
scientific article