Estimates for invariant metrics on $\mathbb C$-convex domains

Nikolov, Nikolai; Pflug, Peter; Zwonek, Włodzimierz

doi:10.1090/s0002-9947-2011-05273-6

Cited by 34 publications

(28 citation statements)

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“…the Lempert theorem, resp. κ D ≤ 4γ D by[11, Corollary 2]. Then Theorem 1 and the estimate D is comparable with γ D and κ D on any non-degenerate C-convex domain D in C n up to multiplicative constants depending only on n.Our second result is about the boundary behavior of the squeezing function near a smooth boundary point of a plane domain.By [2, Theorem 5.3], resp.…”

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confidence: 72%

“…As an application of Theorem 1, we shall prove briefly one of the main results in [11] (whose original proof is close to that of [8, Theorem 1.1]). Denote by γ D , κ D and β D the Carathéodory, the Kobayashi and the Bergman metrics of D. It is well-known that To refine [2, Theorem 5.3], recall that a C 1 -smooth bounded domain D in C n is said to be Dini-smooth if the inner unit normal vector n to ∂D is Dini-continuous.…”

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confidence: 94%

“…Let p ∈ D. We may suppose that p = 0. It follows by [11,Lemma 15] and the proof of [11,Theorem 13] that there exist an orthogonal map T : C n → C n and a lower triangular n × n matrix A with 1's on the main diagonal such that G = T (D) contains the 1-norm unit ball E n in C n and all the coordinates of Az are different from 1 for any point z ∈ G. It is easy to see that ||A|| max ≤ 1 and then ||A −1 || max ≤ (n − 1)!. Denoting by D the unit disc, it follows that the image F n = A(E n ) of E n under the linear map given by A contains 16d n D n , where d n > 0 depends only on n.…”

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confidence: 99%

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Boundary behavior of the squeezing functions of ℂ-convex domains and plane domains

Nikolov

Andreev

2017

Int. J. Math.

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It is shown that any non-degenerate C-convex domain in C n is uniformly squeezing. It is also found the precise behavior of the squeezing function near a Dini-smooth boundary point of a plane domain.Denote by B n the unit ball in C n . Let M be an n-dimensional complex manifold, and p ∈ M. For any holomorphic embedding f :f 's exist, and s M (p) = 0 otherwise. If inf M s M > 0, then M is said to be uniformly squeezing.Many properties and applications of the squeezing function and the uniformly squeezing manifolds have been explored by various authors, see e.g. [2,3,4,5,6,7,8].By [8, Theorem 2.1], any convex bounded domain in C n is uniformly squeezing. Our first aim is to extend this result to a larger class of domains.A domain D in C n is called C-convex if any non-empty intersection of D with a complex line is a simply connected domain. Then C n \ D is a union of hyperplanes (see e.g. [1, Theorem 2.3.9]). This easily implies that if D is degenerate, i.e. containing complex lines, then D is linearly equivalent to C × D ′ , and hence s D = 0.On the other hand, we have the following.2010 Mathematics Subject Classification. 32F45.

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confidence: 72%

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confidence: 94%

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confidence: 99%

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Boundary behavior of the squeezing functions of ℂ-convex domains and plane domains

Nikolov

Andreev

2017

Int. J. Math.

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“…The proof that I ⊂ 2 will be similar to the proof of Proposition 1 in [95]. For X = ϕ (0) ∈Ī by L denote the complex line generated by X .…”

Section: Theorem 74 Let Be Bounded and Hyperconvex Thenmentioning

confidence: 86%

Cauchy–Riemann meet Monge–Ampère

Błocki

2014

Bull. Math. Sci.

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“…Estimates on the infinitesimal Kobayashi metric were established for C-convex sets in [22]. In particular, the Bergman, Carathéodory, and Kobayashi metrics are all bi-Lipschitz [22, Proposition 1, Theorem 12] for C-convex sets which do not contain any complex affine lines.…”

Section: Introductionmentioning

confidence: 99%

Gromov hyperbolicity, the Kobayashi metric, and $\mathbb {C}$-convex sets

Zimmer

2017

Trans. Amer. Math. Soc.

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Abstract. In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to C-convex sets with C 1 -smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded C-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.

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Estimates for invariant metrics on $\mathbb C$-convex domains

Abstract: Abstract. Geometric lower and upper estimates are obtained for invariant metrics on C-convex domains containing no complex lines.

Cited by 34 publications

References 31 publications

Boundary behavior of the squeezing functions of ℂ-convex domains and plane domains

Boundary behavior of the squeezing functions of ℂ-convex domains and plane domains

Cauchy–Riemann meet Monge–Ampère

Gromov hyperbolicity, the Kobayashi metric, and $\mathbb {C}$-convex sets

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