scholarly journals Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains

2018 ◽  
Vol 30 (1) ◽  
pp. 159-170
Author(s):  
Peter Pflug ◽  
Włodzimierz Zwonek
Keyword(s):  
Convex Domain ◽  
Gromov Hyperbolicity ◽  
Smooth Convex ◽  
Tube Domain ◽  
Convex Domains ◽  
Hilbert Metric ◽  
Kobayashi Distance ◽  
Continuity Properties ◽  
Gromov Hyperbolic ◽  
Tube Domains

Abstract We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains. The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain Ω in {\mathbb{R}^{n}} with the Kobayashi distance of the tube domain over the domain Ω is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tube domains with a smooth convex bounded base are also studied in detail.

scholarly journals Zero varieties for the Nevanlinna class on all convex domains of finite type

2001 ◽  
Vol 163 ◽  
pp. 215-227 ◽  
Author(s):  
Klas Diederich ◽  
Emmanuel Mazzilli
Keyword(s):  
Finite Type ◽  
Convex Domain ◽  
Main Tool ◽  
Smooth Convex ◽  
Convex Domains ◽  
Zero Sets ◽  
Nevanlinna Class ◽  
Cauchy Riemann ◽  
Bounded Smooth

It is shown, that the so-called Blaschke condition characterizes in any bounded smooth convex domain of finite type exactly the divisors which are zero sets of functions of the Nevanlinna class on the domain. The main tool is a non-isotropic L1 estimate for solutions of the Cauchy-Riemann equations on such domains, which are obtained by estimating suitable kernels of Berndtsson-Andersson type.

scholarly journals Symmetric homogeneous convex domains

1984 ◽  
Vol 93 ◽  
pp. 1-17
Author(s):  
Tadashi Tsuji
Keyword(s):  
Real Number ◽  
Convex Domain ◽  
Convex Cone ◽  
Affine Transformations ◽  
Convex Domains ◽  
Affine Line ◽  
Homogeneous Convex

Let D be a convex domain in the n-dimensional real number space Rn, not containing any affine line and A(D) the group of all affine transformations of Rn leaving D invariant. If the group A(D) acts transitively on D, then the domain D is said to be homogeneous. From a homogeneous convex domain D in Rn, a homogeneous convex cone V = V(D) in Rn+1 = Rn × R is constructed as follows (cf. Vinberg [11]):which is called the cone fitted on the convex domain D.

SOLUTIONS TO ${\bar\partial}$-EQUATION ON STRONGLY q-CONVEX DOMAINS WITH Lp-ESTIMATES

2004 ◽  
Vol 01 (06) ◽  
pp. 739-749 ◽  
Author(s):  
OSAMA ABDELKADER ◽  
SHABAN KHIDR
Keyword(s):  
Vector Bundle ◽  
Convex Domain ◽  
Holomorphic Vector Bundle ◽  
Kähler Manifold ◽  
Convex Domains ◽  
Lp Estimates ◽  
Type K ◽  
Kahler Manifold

The purpose of this paper is to construct a solution with Lp-estimates, 1≤p≤∞, to the equation [Formula: see text] on strongly q-convex domain of Kähler manifold. This is done for forms of type (n,s), s≥ max (q,k), with values in a holomorphic vector bundle which is Nakano semi-positive of type k and for forms of type (0,s), q≤s≤n-k, with values in a holomorphic vector bundle which is Nakano semi-negative of type k.

Saturated systems of symmetric convex domains; results of Eggleston, Bambah and Woods

1973 ◽  
Vol 74 (1) ◽  
pp. 107-116 ◽  
Author(s):  
Vishwa Chander Dumir ◽  
Dharam Singh Khassa
Keyword(s):  
Lower Bound ◽  
Lebesgue Measure ◽  
Convex Domain ◽  
Gauge Function ◽  
Symmetric Convex ◽  
Convex Domains ◽  
Strictly Convex ◽  
Point Set ◽  
Image Position

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined bywhere V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured thatwhere ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.

scholarly journals The Lindelöf principle and angular derivatives in convex domains of finite type

2002 ◽  
Vol 73 (2) ◽  
pp. 221-250 ◽  
Author(s):  
Marco Abate ◽  
Roberto Tauraso
Keyword(s):  
Finite Type ◽  
Boundary Behaviour ◽  
Convex Domains ◽  
Kobayashi Distance ◽  
Angular Derivatives ◽  
Circular Domains ◽  
Lindelöf Principle ◽  
Real Analytic ◽  
Bounded Convex ◽  
Carathéodory Theorem

AbstractWe describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelöf principle.

scholarly journals APPROXIMATION OF HOLOMORPHIC MAPPINGS ON 1-CONVEX DOMAINS

2013 ◽  
Vol 24 (14) ◽  
pp. 1350108 ◽  
Author(s):  
KRIS STOPAR
Keyword(s):  
Complex Manifold ◽  
Convex Domain ◽  
Holomorphic Mappings ◽  
Holomorphic Maps ◽  
Convex Domains ◽  
Exceptional Set ◽  
Proper Holomorphic Maps ◽  
Pseudoconvex Boundary ◽  
Oka Principle ◽  
Additional Assumptions

Let π : Z → X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D ⋐ X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over [Formula: see text] which has prescribed core, it fixes the exceptional set E of D, and is dominating on [Formula: see text]. Each section in this spray is of class [Formula: see text] and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class [Formula: see text] and holomorphic on D can be approximated in the [Formula: see text] topology by π-sections that are holomorphic in open neighborhoods of [Formula: see text]. Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.

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