Bracci, Filippo; Saracco, Alberto Hyperbolicity in unbounded convex domains. (English) Zbl 1183.32014 Forum Math. 21, No. 5, 815-825 (2009). The authors list various conditions equivalent to (Kobayashi) hyperbolicity for a (possibly unbounded) convex domain \(D\) in \(\mathbb C^n\). In particular, they prove that an unbounded convex domain is hyperbolic if and only if it has global peak and antipeak plurisubharmonic functions at infinity in the sense of H. Gaussier [Proc. Am. Math. Soc 127, No. 1, 105–116 (1999; Zbl 0912.32025)]. The authors also observe that \(D\) can be written, after an appropriate complex affine change of coordinates, as a product \(\mathbb C^k \times D'\) where \(k\) is an integer with \(0 \leq k \leq n\) and \(D'\) is a complete hyperbolic convex domain in \(\mathbb C^{n-k}\); and they give some applications of this fact. Reviewer: Theodore J. Barth (Riverside) Cited in 14 Documents MSC: 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 32A19 Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.) 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32F32 Analytical consequences of geometric convexity (vanishing theorems, etc.) 32F45 Invariant metrics and pseudodistances in several complex variables 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables 32T40 Peak functions 52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces) Keywords:antipeak plurisubharmonic function; Bergman metric; Kobayashi pseudodistance; peak plurisubharmonic function; taut domain; unbounded convex domain Citations:Zbl 0912.32025 PDFBibTeX XMLCite \textit{F. Bracci} and \textit{A. Saracco}, Forum Math. 21, No. 5, 815--825 (2009; Zbl 1183.32014) Full Text: DOI arXiv References: [1] Abate M., Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 pp 167– (4) [2] DOI: 10.2307/2042495 · Zbl 0438.32013 · doi:10.2307/2042495 [3] DOI: 10.1007/s00209-002-0433-7 · Zbl 1018.32021 · doi:10.1007/s00209-002-0433-7 [4] DOI: 10.1090/S0002-9939-99-04492-5 · Zbl 0912.32025 · doi:10.1090/S0002-9939-99-04492-5 [5] Harris, Notase de Matematica 65 pp 345– (1979) [6] DOI: 10.1090/S0002-9904-1970-12363-1 · Zbl 0192.44103 · doi:10.1090/S0002-9904-1970-12363-1 [7] Lempert L., Bull. Soc. Math. France 109 pp 427– (1981) [8] DOI: 10.1007/BF02201775 · Zbl 0509.32015 · doi:10.1007/BF02201775 [9] DOI: 10.1090/S0002-9939-03-07030-8 · Zbl 1020.32001 · doi:10.1090/S0002-9939-03-07030-8 [10] DOI: 10.4064/ap81-1-6 · Zbl 1022.32001 · doi:10.4064/ap81-1-6 [11] Nikolov N., Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 pp 601– (5) [12] DOI: 10.1007/BFb0058768 · doi:10.1007/BFb0058768 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.