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

Bracci, Filippo; Gaussier, Hervé; Zimmer, Andrew

doi:10.1007/s00208-020-01954-1

Cited by 17 publications

(21 citation statements)

References 34 publications

(46 reference statements)

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“…Theorem 1.2 is a consequence of Corollary 4.5 and Theorem 5.3. We remark that the equivalence between Gromov and euclidean compactifications had already been proven in [10,42] for cases (1) and (3), and in case (2) this equivalence follows immediately from results in [13,45]. Notice also that since the horofunction and the Gromov compactifications are metric invariants, the equivalence between D G and D H also holds for every domain of C q which is biholomorphic to the ones in (1),( 2),(3).…”

supporting

confidence: 52%

The horofunction boundary of a Gromov hyperbolic space

Arosio¹,

Fiacchi²,

Gontard³

et al. 2020

Preprint

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We highlight a condition, the approaching geodesics property, on a proper geodesic Gromov hyperbolic metric space, which implies that the horofunction compactification is topologically equivalent to the Gromov compactification. It is known that this equivalence does not hold in general. We prove using rescaling techniques that the approaching geodesics property is satisfied by bounded strongly pseudoconvex domains of C q endowed with the Kobayashi metric. We also show that bounded convex domains of C q with boundary of finite type in the sense of D'Angelo satisfy a weaker property, which still implies the equivalence of the two said compactifications. As a consequence we prove that on those domains big and small horospheres as defined by Abate in [1] coincide. Finally we generalize the classical Julia's lemma, giving applications to the dynamics of non-expanding maps. Contents Non-expanding self-maps 19References 29

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confidence: 52%

The horofunction boundary of a Gromov hyperbolic space

Arosio¹,

Fiacchi²,

Gontard³

et al. 2020

Preprint

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“…Thus the visual metric of ∂ G Ω and the Euclidean metric of ∂Ω are bi-Hölder equivalent to each other. On the other hand, Bracci, Gaussier and Zimmer [7] get the following result on convex domains. Theorem 1.10 (Theorem 1.4, [7]).…”

Section: Introductionmentioning

confidence: 95%

“…On the other hand, Bracci, Gaussier and Zimmer [7] get the following result on convex domains. Theorem 1.10 (Theorem 1.4, [7]). Let Ω be a C-proper convex domain on C n .…”

Section: Introductionmentioning

confidence: 95%

The Gehring-Hayman type theorems on complex domains

Liu,

Wang,

Zhou

2020

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In this paper we establish Gehring-Hayman type theorems for some complex domains. Suppose that Ω ⊂ C n is a bounded m-convex domain with Dini-smooth boundary, or a bounded strongly pseudoconvex domain with C 2 -smooth boundary. Then we prove that the Euclidean length of Kobayashi geodesic [x, y] in Ω is less than c 1 |x − y| c 2 . Furthermore, if Ω endowed with the Kobayashi metric is Gromov hyperbolic, then we can generalize this result to quasi-geodesics with respect to Bergman metric, Carathéodory metric or Kähler-Einstein metric.As applications, we prove the bi-Hölder equivalence between the Euclidean boundary and the Gromov boundary. Moreover, by using this boundary correspondence, we can show some extension results for biholomorphisms, and more general rough quasi-isometries with respect to the Kobayashi metrics between the domains.Theorem 1.2. Let Ω be a bounded m-convex domain in C n (n ≥ 2) with Dinismooth boundary. Then for any 0 < c 2 < 1/(12m 2 − 8m), there exists a constant c 1 > 0 such that, for any x, y ∈ Ω,where [x, y] is any Kobayashi geodesic joining x and y in Ω.If, in addition, (Ω, K Ω ) is Gromov hyperbolic and γ is a Kobayashi λ-quasigeodesic connecting x and y with λ ≥ 1, then there exists a constant c ′ 1 > 0 such thatRecently Zimmer showed the following theorem.The following result is a direct consequence of Theorem 1.2 and Theorem 1.3.Corollary 1.4. Suppose that Ω is a bounded convex domain in C n (n ≥ 2) with Dini-smooth boundary and that (Ω, K Ω ) is Gromov hyperbolic, where K Ω is the Kobayashi metric of Ω. Then for any λ ≥ 1, there exist constants c 1 , c 2 > 0 such that for all x, y ∈ Ω,where γ is a λ-quasi-geodesic in the Kobayashi metric with end points x and y.These results show that the Kobayashi geodesics (or quasi-geodesics) are essentially also short in the Euclidean sense.The proof of Theorem 1.2 requires the following lemma. In what follows, denote a ∨ b := max{a, b} and a ∧ b := min{a, b} for a, b ∈ R.

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“…In [6,7,8] this point of view has been used to prove extension of biholomorphisms between Gromov hyperbolic convex domains, proving, for instance, that every convex map from the ball whose image is convex extends as a homeomorphism up to the boundary regardless the regularity of the image. In [20,21] it has been proved that Gromov hyperbolicity of convex smooth bounded domains is related to D'Angelo type finiteness of the boundary, while in [15] the same result has been proved in C 2 for pseudoconvex domains.…”

Section: Introductionmentioning

confidence: 99%

“…(1) bounded smooth strongly pseudoconvex domains (see [1]), (2) bounded smooth convex domains of finite D'Angelo type (see [20]), (3) Gromov hyperbolic (with respect to the Kobayashi distance) convex domains (see [8]), (4) bounded smooth pseudoconvex domains of finite D'Angelo type in C 2 (see [15]), (5) bounded Gromov hyperbolic (with respect to the Kobayashi distance) C-convex domains with Lipschitz boundary (see [21] for the C 1 -smooth case and Section 4 for the Lipschitz case) (6) any domain biholomorphic to a Gromov model domain such that the biholomorphism extends as a homeomorphism up to the boundary. Theorem 1.6 allows to "localize" the previous list as follows:…”

Section: Introductionmentioning

confidence: 99%

Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance

Bracci¹,

Gaussier²,

Nikolov³

et al. 2022

Preprint

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We introduce the notion of locally visible and locally Gromov hyperbolic domains in C d . We prove that a bounded domain in C d is locally visible and locally Gromov hyperbolic if and only if it is (globally) visible and Gromov hyperbolic with respect to the Kobayashi distance. This allows to construct new classes of domains which are Gromov hyperbolic and for which biholomorphisms extend continuously up to the boundary.

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

Cited by 17 publications

References 34 publications

The horofunction boundary of a Gromov hyperbolic space

The horofunction boundary of a Gromov hyperbolic space

The Gehring-Hayman type theorems on complex domains

Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance

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