Chern, Shiing-Shen; Ji, Shanyu On the Riemann mapping theorem. (English) Zbl 0872.32016 Ann. Math. (2) 144, No. 2, 421-439 (1996). The present paper is a continuation of the authors’ previous work concerning projective geometry and Riemann’s mapping problem [S. S. Chern and S. Ji, Math. Ann. 302, No. 3, 581-600 (1995; Zbl 0843.32013)]. The main result is the following theorem (a generalization of the Riemann mapping theorem): If a bounded simply connected domain \(\Omega\) in \(\mathbb{C}^n\) has connected, smooth, (locally) spherical boundary, then it is biholomorphic to the ball. The boundary is called (locally) spherical if it is locally CR-equivalent to a portion of the unit sphere.The authors point out that when the boundary of \(\Omega\) is simply connected, the result also follows from earlier work of Pinčuk [S. Pinčuk, Math. USSR Sb. 27(1975), 375-392 (1977); translation from Mat. Sb, n. Ser. 98(140), 416-435 (1975; Zbl 0366.32010)]. Reviewer: E.J.Straube (College Station) Cited in 3 ReviewsCited in 11 Documents MSC: 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 53C56 Other complex differential geometry Keywords:Segre family; projective structure bundle; connections; Riemann mapping theorem Citations:Zbl 0843.32013; Zbl 0366.32010 PDFBibTeX XMLCite \textit{S.-S. Chern} and \textit{S. Ji}, Ann. Math. (2) 144, No. 2, 421--439 (1996; Zbl 0872.32016) Full Text: DOI