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Gromov (non-)hyperbolicity of certain domains in ℂN

  • Nikolai Nikolov , Pascal J. Thomas EMAIL logo and Maria Trybuła
Published/Copyright: July 21, 2015
Forum Mathematicum
From the journal Forum Mathematicum Volume 28 Issue 4

Abstract

We prove the Gromov non-hyperbolicity with respect to the Kobayashi distance for 𝒞1,1-smooth convex domains in ℂ2 which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. The same is shown for “generic” product spaces, as well as for the symmetrized polydisc and the tetrablock. On the other hand, examples of smooth, non-pseudoconvex, Gromov hyperbolic domains in ℂn are given.

Keywords: Convex domains; Kobayashi distance; Gromov hyperbolicity
MSC 2010: 32F17; 32F45

Funding statement: Research of the third author is supported by the International PhD programme “Geometry and Topology in Physical Models” of the Foundation for Polish Science, and by the Bulgarian National Science Found (contract DFNI-I 02/14). The initial version of this paper was prepared during her visit to the Institute of Mathematics and Informatics, Bulgarian Academy of Science, October 2013–April 2014.

A Appendix

Proposition A.1

Proposition A.1 (cf. [16, Proposition 2])

The following statements hold.

  1. Let D be proper convex domain in ℂn. Then

    cD⁢(z,w)≥12⁢|log⁡dD⁢(z)dD⁢(w)|,z,w∈D.

  2. Let D be proper ℂ-convex domain in ℂn. Then

    cD⁢(z,w)≥14⁢|log⁡dD⁢(z)dD⁢(w)|,z,w∈D.

Proposition A.2

Proposition A.2 (see [11, proof of Proposition 10.2.3])

Let b be a C1,1-smooth boundary point of a domain D is Cn and let K⋐D. Then there exist a neighborhood U of b and a constant C>0 such that

2⁢kD⁢(z,w)≤-log⁡dD⁢(z)+C,z∈D∩U,w∈K.
Proposition A.3

Proposition A.3 (cf. [7, Theorem 1])

Let b be a C2-smooth non-pseudoconvex boundary point of a domain D in C2. Then there exist a neighborhood U of b and a constant c>0 such that

c⁢κD⁢(z;X)≤|〈∇⁡dD⁢(z),X〉|(dD⁢(z))34+|X|,z∈D∩U,X∈ℂn.
Proposition A.4

Let D be a bounded domain in Cn. Let U and V be neighborhoods of ∂⁡D with V⋐U. Then there exists a constant c>0 such that for any connected component D′ of D∩U one has that

c⁢kD′⁢(z,w)≤kD⁢(z,w),z,w∈D′∩V.

Proof.

Let ε>0. Take a smooth curve γ:[0,1]→D such that γ⁢(0)=z, γ⁢(1)=w and

kD⁢(z,w,ε):=kD⁢(z,w)+ε>∫01κD⁢(γ⁢(t);γ′⁢(t))⁢𝑑t.

Let s=sup⁡{t∈(0,1):γ⁢(0,t)⊂D′∩V} and r=inf⁡{t≥s:γ⁢([t,1])⊂D′∩V}. Set z′=γ⁢(s) and w′=γ⁢(r). The localization property of the Kobayashi metric (cf. [11, Proposition 7.2.9], or [8]) provides a constant c′>0 such that

c′⁢κD′⁢(u;X)≤κD⁢(u;X),z∈D′∩V,X∈ℂn.

It follows that

kD⁢(z,w,ε)>c′⁢kD′⁢(z,z′)+kD⁢(z′,w′)+c′⁢kD′⁢(w′,w)≥c′⁢kD′⁢(z,w)+kD⁢(z′,w′)-c′⁢kD′⁢(z′,w′).

If z′≠w′, then z′,w′∈D′∩∂⁡V⋐D′. Then there exists a constant c1>0 such that

kD′⁢(u,v)≤c1⁢∥u-v∥,u,v∈D′∩∂⁡V.

On the other hand, since D is bounded, we may find a constant c2>0 such that

kD⁢(u,v)≥c2⁢∥u-v∥,u,v∈D′∩∂⁡V.

Then

kD⁢(z,w,ε)>c′⁢kD′⁢(z,w)+(c2-c′⁢c1)⁢∥z′-w′∥.

Since

kD⁢(z,w,ε)>kD⁢(z′,w′)≥c2⁢∥z′-w′∥,

we get that

kD⁢(z,w,ε)>c′⁢kD′⁢(z,w)-(c′⁢c1c2-1)+⁢kD⁢(z,w,ε).

The last inequality also holds if z′=w′. Letting ε→0, we obtain that

kD⁢(z,w)≥min⁡{c′,c2c1}⁢kD′⁢(z,w).∎

The authors would like to thank the referee for his/her valuable comments.

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Received: 2014-6-24
Revised: 2015-3-19
Published Online: 2015-7-21
Published in Print: 2016-7-1

© 2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Residues of standard intertwining operators on p-adic classical groups
  3. Anomalies on codimension growth of algebras
  4. Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras
  5. Bounded gaps between primes with a given primitive root, II
  6. Minimal potential results for the Schrödinger equation in a slab
  7. The classification of p-nilpotent restricted Lie algebras of dimension at most 4
  8. Global gradient estimates for nonlinear obstacle problems with nonstandard growth
  9. Existence of the Bedrosian identity for Fourier multiplier operators
  10. Projective bundles over small covers and the bundle triviality problem
  11. Gromov (non-)hyperbolicity of certain domains in ℂN
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  13. Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian
https://doi.org/10.1515/forum-2014-0113
Keywords for this article
Convex domains; Kobayashi distance; Gromov hyperbolicity

Articles in the same Issue

  1. Frontmatter
  2. Residues of standard intertwining operators on p-adic classical groups
  3. Anomalies on codimension growth of algebras
  4. Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras
  5. Bounded gaps between primes with a given primitive root, II
  6. Minimal potential results for the Schrödinger equation in a slab
  7. The classification of p-nilpotent restricted Lie algebras of dimension at most 4
  8. Global gradient estimates for nonlinear obstacle problems with nonstandard growth
  9. Existence of the Bedrosian identity for Fourier multiplier operators
  10. Projective bundles over small covers and the bundle triviality problem
  11. Gromov (non-)hyperbolicity of certain domains in ℂN
  12. The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}
  13. Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian
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