On isometries of the Kobayashi and Carathéodory metrics

Mahajan, Prachi

doi:10.4064/ap104-2-2

Cited by 5 publications

(7 citation statements)

References 29 publications

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“…If D ⊂ C n is a bounded strongly convex domain with C 3 boundary, and X ⊂ C m and Y ⊂ C k are bounded convex domains, then (X ×Y, k X×Y ) cannot be isometrically embedded into (D, k D ). These theorems extend results by Bracci and Gaussier [5], Zwonek [19], and resolves a question by Mahajan [13], see also [6].…”

Section: Introductionsupporting

confidence: 85%

A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains

Lemmens¹

2020

Preprint

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We show that If D ⊂ C n is a bounded strongly convex domain with C 3 boundary, and X ⊂ C m and Y ⊂ C k are bounded convex domains, then X × Y cannot be isometrically embedded into D under the Kobayashi distance. This result generalises Poincaré's theorem which says that there is no biholomorphic map from the polydisc onto the (open) Euclidean ball in C n for n ≥ 2.The method of proof only relies on the metric geometry of the spaces and will be derived from a result for products of proper geodesic metric spaces. In fact, we prove that if M 1 , M 2 , and N are proper geodesic metric spaces, where both M 1 and M 2 contain an almostgeodesic ray, and for any two distinct Busemann points in the metric compactification of N the detour distance is infinite, then the product metric space M 1 × M 2 with the product distance cannot be isometrically embedded into N .

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Section: Introductionsupporting

confidence: 85%

A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains

Lemmens¹

2020

Preprint

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“…Hence the result. Finally, we recall a localization result (Lemma 4.3 of [39]) for the Kobayashi balls. The proof relies on the localization property of the Kobayashi metric near holomorphic peak points (see [25], [47]).…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

“…This is useful in proving the isometric inequivalence of strictly weakly spherical Levi corank one domains (the notion of weak sphericity is recalled from [7] and defined in the last section) and strongly pseudoconvex domains -see [39] for a related result in C 2 .…”

Section: Setmentioning

confidence: 99%

“…However, this seems to be unknown for isometries of the Kobayashi or the Carathéodory metric between a pair of strongly pseudoconvex domains. On the other hand, the results of [39] and [6] show that isometries behave very much like holomorphic mappings. In particular, they exhibit essentially the same boundary behaviour as biholomorphisms.…”

Section: Introductionmentioning

confidence: 98%

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Bounds for invariant distances on pseudoconvex Levi corank one domains and applications

Balakumar¹,

Mahajan²,

Verma³

2015

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Let D ⊂ C n be a smoothly bounded pseudoconvex Levi corank one domain with defining function r, i.e., the Levi form ∂∂r of the boundary ∂D has at least (n − 2) positive eigenvalues everywhere on ∂D. The main goal of this article is to obtain bounds for the Carathéodory, Kobayashi and the Bergman distance between a given pair of points p, q ∈ D in terms of parameters that reflect the Levi geometry of ∂D and the distance of these points to the boundary. Applications include an understanding of Fridman's invariant for the Kobayashi metric on Levi corank one domains, a description of the balls in the Kobayashi metric on such domains that are centered at points close to the boundary in terms of Euclidean data and the boundary behaviour of Kobayashi isometries from such domains.

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“…In this paper, we shall see that a classic theorem due to Poincaré [22], which says that there is no biholomorphic map from the polydisc n onto the (open) Euclidean ball B n in C n if n ≥ 2, is a case in point. In fact, it is known [19,29,30] that there exists no surjective Kobayashi distance isometry of n onto B n if n ≥ 2. More…”

Section: Introductionmentioning

confidence: 99%

A Metric Version of Poincaré’s Theorem Concerning Biholomorphic Inequivalence of Domains

Lemmens

2022

J Geom Anal

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We show that if $$Y_j\subset \mathbb {C}^{n_j}$$ Y j ⊂ C n j is a bounded strongly convex domain with $$C^3$$ C 3 -boundary for $$j=1,\dots ,q$$ j = 1 , ⋯ , q , and $$X_j\subset \mathbb {C}^{m_j}$$ X j ⊂ C m j is a bounded convex domain for $$j=1,\ldots ,p$$ j = 1 , … , p , then the product domain $$\prod _{j=1}^p X_j\subset \mathbb {C}^m$$ ∏ j = 1 p X j ⊂ C m cannot be isometrically embedded into $$\prod _{j=1}^q Y_j\subset \mathbb {C}^n$$ ∏ j = 1 q Y j ⊂ C n under the Kobayashi distance, if $$p>q$$ p > q . This result generalises Poincaré’s theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in $$\mathbb {C}^n$$ C n for $$n\ge 2$$ n ≥ 2 . The method of proof only relies on the metric geometry of the spaces and will be derived from a more general result for products of proper geodesic metric spaces with the sup-metric. In fact, the main goal of the paper is to establish a general criterion, in terms of certain asymptotic geometric properties of the individual metric spaces, that yields an obstruction for the existence of an isometric embedding between product metric spaces.

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On isometries of the Kobayashi and Carathéodory metrics

Cited by 5 publications

References 29 publications

A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains

A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains

Bounds for invariant distances on pseudoconvex Levi corank one domains and applications

A Metric Version of Poincaré’s Theorem Concerning Biholomorphic Inequivalence of Domains

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