Hyperbolicity in unbounded convex domains

Bracci, Filippo; Saracco, Alberto

doi:10.1515/forum.2009.039

Cited by 26 publications

(23 citation statements)

References 13 publications

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“…On the other hand, if D ⊂ C d is a hyperbolic convex domain and f : D → D is holomorphic, then f has no fixed points in D if and only if every orbit of {f n } is compactly divergent (see [1,2,35,17]). Therefore, as a direct corollary to Karlsson's result and Theorem 1.4 we have the following:…”

Section: Introduction and Resultsmentioning

confidence: 99%

Homeomorphic extension of quasi-isometries for convex domains in $${\mathbb {C}}^d$$ and iteration theory

2020

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We study the homeomorphic extension of biholomorphisms between convex domains in C d without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov boundary and the topological boundaries of the domains and the dynamical properties of commuting 1-Lipschitz maps in Gromov hyperbolic spaces. This approach not only allows us to prove extensions for biholomorphisms, but for more general quasi-isometries between the domains endowed with their Kobayashi distances.

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Section: Introduction and Resultsmentioning

confidence: 99%

Homeomorphic extension of quasi-isometries for convex domains in $${\mathbb {C}}^d$$ and iteration theory

2020

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“…By [13], a (possibly unbounded) convex domain in C N is complete hyperbolic if and only if it is biholomorphic to a bounded domain of C N , in particular, a convex domain is hyperbolic if and only if it is complete hyperbolic.…”

Section: Convex Domainsmentioning

confidence: 99%

“…Since D is a complete hyperbolic convex domain, for every j there exists a complex geodesic ϕ j : D → D such that ϕ j (0) = p j and ϕ j (t j ) = q j for some t j ∈ (0, 1), see [13,Lemma 3.3]. By (6.8) it follows…”

Section: 2mentioning

confidence: 99%

Horosphere topology

Bracci¹,

Gaussier²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We introduce a prime end-type theory on complete Kobayashi hyperbolic manifolds using horosphere sequences. This allows to introduce a new notion of boundary-new even in the unit disc in the complex space-the horosphere boundary, and a topology on the manifold together with its horosphere boundary, the horosphere topology. We prove that a bounded strongly pseudoconvex domain endowed with the horosphere topology is homeomorphic to its Euclidean closure, while for the polydisc such a horosphere topology is not even Hausdorff and is different from the Gromov topology. We use this theory to study boundary behavior of univalent maps from bounded strongly pseudoconvex domains.

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“…If D is a bounded domain then D is always Kobayashi hyperbolic. In the case D is a convex domain then we may easily get that D is linearly isomorphic with C k × D ′ , where 0 ≤ k ≤ n and D ′ ⊂ C n−k is a Kobayashi hyperbolic convex domain (see e. g. Proposition 1.2 in [5]).…”

Section: Introductionmentioning

confidence: 99%

Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains

Pflug

Zwonek

2017

Forum Mathematicum

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We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain Ω in R n with the Kobayashi distance of the tube domain over the domain Ω is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tube domains with a smooth convex bounded base are also studied in detail.2010 Mathematics Subject Classification. 32A07, 32F45, 53C25.

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Hyperbolicity in unbounded convex domains

Abstract: ABSTRACT. We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.

Cited by 26 publications

References 13 publications

Homeomorphic extension of quasi-isometries for convex domains in $${\mathbb {C}}^d$$ and iteration theory

Homeomorphic extension of quasi-isometries for convex domains in $${\mathbb {C}}^d$$ and iteration theory

Horosphere topology

Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains

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