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Hyperbolicity in unbounded convex domains. (English) Zbl 1183.32014

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.

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)

Citations:

Zbl 0912.32025
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References:

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