Some Aspects of the Kobayashi and Carathéodory Metrics on Pseudoconvex Domains

Mahajan, Prachi; Verma, Kaushal

doi:10.1007/s12220-010-9206-4

Cited by 14 publications

(24 citation statements)

References 34 publications

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“…the bounds K and C given there can be taken independently of t ∈ T and of ζ, ξ ∈ ∂G t -in the first case depending only on ζ − ξ (with the domains G t not necessarily strictly pseudoconvex, when it comes to the upper estimate), see Propositions 3.5 and 3.7 below. These results are inspired by Propositions 9.1 and 9.2 from [10]. In correspondence to that paper, note that the role of the set of parameters T is there played by a convergent sequence of numbers with its limit added.…”

Section: Introductionmentioning

confidence: 52%

Peak functions and boundary behaviour of holomorphically invariant distances and metrics on strictly pseudoconvex domains

Lewandowski¹

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We give a parameter version of Graham-Kerzman approximation theorem for bounded holomorphic functions on strictly pseudoconvex domains. As an application, we present some uniform estimates for the boundary behaviour of the Kobayashi and Carathéodory pseudodistences on such domains.2010 Mathematics Subject Classification. Primary 32T40; Secondary 32T15, 32F45. Key words and phrases. strictly pseudoconvex domains, peak functions, Kobayashi pseudodistance, Carathéodory pseudodistance.Moreover, if m = 1, then N can be chosen to be zero, (C) f t − f t Gt∩B(ζt,ρ) < ε f Gt∩B(ζt,R) .Remark 1.6. Notice that if m = 1, then the estimate in (B) depends in fact only on ε and R.

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Section: Introductionmentioning

confidence: 52%

Peak functions and boundary behaviour of holomorphically invariant distances and metrics on strictly pseudoconvex domains

Lewandowski¹

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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“…Analogues of this for weakly pseudoconvex finite type domains in C 2 and convex finite type domains in C n were obtained in [41] by a direct scaling. These estimates are useful in establishing a generalized sub-mean value property for plurisubharmonic functions and defining suitable approach regions for boundary values of functions in H p spaces at least on strongly pseudoconvex domains (see [36] for example).…”

Section: Setmentioning

confidence: 69%

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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“…Note that D supports a local holomorphic peak function at p 0 since the boundary ∂D is C 2smooth near it and hence the proof of Theorem 1.1 of [9] can be adapted to show that…”

Section: Two Examplesmentioning

confidence: 99%

“…While both (i) and (ii) use the methods of scaling, they rely on an observation made in [9] namely, the convergence of the integrated Kobayashi distance on each scaled domain to that in the limiting domain. More specifically, refer to Lemma 5.2 and 5.7 of [9].…”

Section: Introductionmentioning

confidence: 99%