Visibility of Kobayashi geodesics in convex domains and related properties

Bracci, Filippo; Nikolov, Nikolai; Thomas, Pascal J.

doi:10.1007/s00209-022-02978-w

Cited by 20 publications

(9 citation statements)

References 33 publications

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“…This generalizes [2, Theorem 1.1]. The idea of the proof is similar to the proof of Theorem 1.1 in [2] along with a few important new observations. Theorem 0.1.…”

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confidence: 59%

Localization of the Kobayashi distance for any visibility domain

Sarkar¹

2022

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In this article, we prove a localization result for the Kobayashi distance for Kobayashi hyperbolic domains in the complex Euclidean space.In this article, we prove the following localization results for the Kobayashi distance of any visibility domain. This generalizes [2, Theorem 1.1]. The idea of the proof is similar to the proof of Theorem 1.1 in [2] along with a few important new observations. Theorem 0.1. Suppose Ω ⊂ C d is a hyperbolic domain and U an open subset of C d such that U ∩ ∂Ω = ∅ and U ∩ Ω is connected. Suppose (U ∩ Ω, k U ∩Ω ) has weak-visibility property for every pair of distinct points in U ∩ ∂Ω. Then for every W ⊂⊂ U , there exists a constant C > 0 which depends only on U, W such that for every z, w ∈ W ∩ Ω,has weak-visibility property for every pair of distinct points in U ∩ ∂Ω. Then for every W ⊂⊂ U , there exists a constant C > 0 which depends only on U, W such that for every z, w ∈ W ∩ Ω,In the above two theorems and throughout, we are assuming that cardinality of U ∩∂Ω is greater than 1; when the cardinality of U ∩ ∂Ω is 1, the localization results above follow trivially. Suppose Ω, U and W as in the above theorems. In this article, we also assume that inf z∈W k Ω (z, U c ) > 0. We shall see later that this condition is a necessary condition for weak-visibility.The following localization lemma, by L. H. Royden [7, Lemma 2], whose proof can be found in [5, Lemma 4], used to prove localization of the Kobayshi distance and to study the relation betwen local and global visibility and Gromov hyperbolicity in [2] and [1] respectively.Next, we prove a lemma which is used to prove the localization of the Kobayshi distance for visibility domains stated above. Lemma 0.3. Suppose Ω ⊂ C d be a Kobayashi hyperbolic domain and U is an open subset of C d such that U ∩ Ω = ∅ and connected. Then for every W ⊂⊂ U , there eixts L > 0 such that for all z ∈ W ∩ Ω and v ∈ C d , (0.2)κ U ∩Ω (z; v) ≤ 1 + Le −k Ω (z,U c ) κ Ω (z; v).

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“…This generalizes [2, Theorem 1.1]. The idea of the proof is similar to the proof of Theorem 1.1 in [2] along with a few important new observations. Theorem 0.1.…”

supporting

confidence: 59%

Localization of the Kobayashi distance for any visibility domain

Sarkar¹

2022

Preprint

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“…We see that the two key properties that the pair (Ω, q) must possess, in addition to the issue of the regularity ∂Ω near q that was discussed in Section 1, that make the above proof work are (i) a localization result for K Ω akin to Result 5.6, and (ii) whatever that leads to the estimate (5.3). Readers familiar with the results in [4] might notice that a weaker condition than O ∩ Ω being log-type convex provides just these two ingredients. That said:…”

Section: The Proof Of Theorem 18mentioning

confidence: 99%

“…It is for these reasons that we highlight Theorem 1.8 in Section 1. Also, while some conditions are now known that will ensure that O ∩ Ω is convex and has the visibility property (i.e., the property introduced in [4]), the latter property is slightly non-explicit. Moreover, the latter is not the last word on properties that will produce a localization result akin to Result 5.6; this will be discussed in forthcoming work.…”

Section: The Proof Of Theorem 18mentioning

confidence: 99%

“…Since the proof of the above is very similar to the argument in Step 1 of the proof in Section 7, we only mention the differences in the argument. Firstly, the role of Result 5.6 is played by [4,Theorem 1.4]. Secondly, the visibility property of O ∩ Ω ensures that (5.3) is true.…”

Section: The Proof Of Theorem 18mentioning

confidence: 99%

“…Readers familiar with the essence of the argument of Forstnerič-Rosay in [9] and with the results in [4] by Bracci et al might ask whether the conclusions of Theorem 1.8 may be obtainable under weaker conditions on O ∩ Ω. We shall address this in Section 7: see Remark 7.1 and Theorem 7.2.…”

Section: Introductionmentioning

confidence: 97%

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Local continuous extension of proper holomorphic maps: low-regularity and infinite-type boundaries

Banik¹

2022

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We prove a couple of results on local continuous extension of proper holomorphic maps F : D → Ω, D, Ω C n , making local assumptions on ∂D and ∂Ω. The first result allows us to have much lower regularity, for the patches of ∂D, ∂Ω that are relevant, than in earlier results. The second result (and a result closely related to it) is in the spirit of a result by Forstnerič-Rosay. However, our assumptions allow ∂Ω to contain boundary points of infinite type.

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