Comparison Between a Gromov Hyperbolic Metric and the Hyperbolic Metric

2018 ◽  
Vol 18 (4) ◽  
pp. 717-722 ◽  
Author(s):  
Xiaohui Zhang
Keyword(s):  
Hyperbolic Metric ◽  
Gromov Hyperbolic ◽  
The Hyperbolic Metric

scholarly journals Diastatic entropy and rigidity of complex hyperbolic manifolds

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Roberto Mossa
Keyword(s):  
Lower Bound ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Metric ◽  
Continuous Map ◽  
The Hyperbolic Metric ◽  
Geometric Rigidity ◽  
Volume Entropy ◽  
Complex Hyperbolic Manifold ◽  
Real Analytic

AbstractLet f : Y → X be a continuous map between a compact real analytic Kähler manifold (Y, g) and a compact complex hyperbolic manifold (X, g0). In this paper we give a lower bound of the diastatic entropy of (Y, g) in terms of the diastatic entropy of (X, g0) and the degree of f . When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary,when X = Y,we get that the minimal diastatic entropy is achieved if and only if g is isometric to the hyperbolic metric g0.

scholarly journals The Harmonic Mapping Whose Hopf Differential Is a Constant

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1310
Author(s):  
Liang Shen
Keyword(s):  
Unit Disk ◽  
Harmonic Mapping ◽  
Hyperbolic Metric ◽  
Beltrami Coefficient ◽  
Hopf Differential ◽  
The Hyperbolic Metric

Suppose that h(z) is a harmonic mapping from the unit disk D to itself with respect to the hyperbolic metric. If the Hopf differential of h(z) is a constant c>0, the Beltrami coefficient μ(z) of h(z) is radially symmetric and takes the maximum at z=0. Furthermore, the mapping γ:c→μ(0) is increasing and gives a homeomorphism from (0,+∞) to (0,1).

Normal Functions: Lp Estimates

1997 ◽  
Vol 49 (1) ◽  
pp. 55-73 ◽  
Author(s):  
Huaihui Chen ◽  
Paul M. Gauthier
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Holomorphic Functions ◽  
Hyperbolic Metric ◽  
Lp Estimates ◽  
Normal Functions ◽  
The Hyperbolic Metric

AbstractFor ameromorphic (or harmonic) function ƒ, let us call the dilation of ƒ at z the ratio of the (spherical)metric at ƒ(z) and the (hyperbolic)metric at z. Inequalities are knownwhich estimate the sup norm of the dilation in terms of its Lp norm, for p > 2, while capitalizing on the symmetries of ƒ. In the present paper we weaken the hypothesis by showing that such estimates persist even if the Lp norms are taken only over the set of z on which ƒ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that p = 2.

Hyperbolic Metric and Multiply Connected Wandering Domains of Meromorphic Functions

2017 ◽  
Vol 60 (3) ◽  
pp. 787-810
Author(s):  
Jian-Hua Zheng
Keyword(s):  
Analytic Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Hyperbolic Metric ◽  
Multiply Connected ◽  
The Hyperbolic Metric ◽  
Large Round ◽  
Hyperbolic Domain ◽  
Wandering Domains

AbstractIn this paper, in terms of the hyperbolic metric, we give a condition under which the image of a hyperbolic domain of an analytic function contains a round annulus centred at the origin. From this, we establish results on the multiply connected wandering domains of a meromorphic function that contain large round annuli centred at the origin. We thereby successfully extend the results of transcendental meromorphic functions with finitely many poles to those with infinitely many poles.

Sign in / Sign up

Export Citation Format

Share Document