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Register a new userOn the Gromov hyperbolicity of convex domains in Cn
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Herve Gaussier
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We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi distance. We also provide examples of bounded smooth convex domains that are not strongly pseudoconvex but are Gromov hyperbolic.
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In this paper we study the global geometry of the Kobayashi metric on convex sets. We provide new examples of non-Gromov hyperbolic domains in $mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon ewline -vex, bounded and unbounded. Our first aim is to prove that if $Omega$ is a bounded weakly linearly convex domain in $mathbb{C}^n,,ngeq 2,$ and $S$ is an affine complex hyperplane intersecting $Omega,$ then the domain $Omegasetminus S$ endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next we localize this result on Kobayashi hyperbolic convex domains. Namely, we show that Gromov hyperbolicity of every open set of the form $Omegasetminus S,$ where $S$ is relatively closed in $Omega$ and $Omega$ is a convex domain, depends only on that how $S$ looks near the boundary, i.e., whether $S$ near $partialOmega$ (Theorem 1.7). We close the paper with a general remark on Hartogs type domains. The paper extends in an essential way results in [6].
We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows to deduce the non-hyperbolicity of any periodic Denjoy domain.
We study the homeomorphic extension of biholomorphisms between convex domains in $mathbb C^d$ 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.
If $X$ is a geodesic metric space and $x_{1},x_{2},x_{3} in X$, a geodesic triangle $T={x_{1},x_{2},x_{3}}$ is the union of the three geodesics $[x_{1}x_{2}]$, $[x_{2}x_{3}]$ and $[x_{3}x_{1}]$ in $X$. The space $X$ is $delta$-hyperbolic in the Gromov sense if any side of $T$ is contained in a $delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $delta(X) =inf { deltageq 0:{0.3cm}$ X ${0.2cm}$ $text{is} {0.2cm} delta text{-hyperbolic} }.$ To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolycity constant in terms of some parameters of the graph. Denote by $mathcal{G}(n,m)$ the set of graphs $G$ with $n$ vertices and $m$ edges, and such that every edge has length $1$. In this work we estimate $A(n,m):=min{delta(G)mid G in mathcal{G}(n,m) }$ and $B(n,m):=max{delta(G)mid G in mathcal{G}(n,m) }$. In particular, we obtain good bounds for $B(n,m)$, and we compute the precise value of $A(n,m)$ for all values of $n$ and $m$. Besides, we apply these results to random graphs.
The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $mathbb R^n$, $nge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashis pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $mathbb R^n$, this minimal metric coincides with the classical Beltrami-Cayley-Klein metric. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a domain having a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric. Finally, we show that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-plane.
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