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).

A Filter Description of the Homomorphisms of H∞

1965 ◽  
Vol 17 ◽  
pp. 734-757 ◽  
Author(s):  
A. Kerr-Lawson
Keyword(s):  
Unit Disk ◽  
Maximal Ideal ◽  
Continuous Mapping ◽  
Open Unit ◽  
Supremum Norm ◽  
Ideal Space ◽  
Open Unit Disk ◽  
Classical Theorem ◽  
Bounded Analytic Functions ◽  
The Hyperbolic Metric

The algebra of bounded analytic functions on the open unit disk D, usually written H∞ i is a commutative Banach algebra under the supremum norm. Since its compact maximal ideal space M (space of complex homomorphisms) is an extension space of the unit disk, there must be a continuous mapping form βD, the Stone-Čech compactification of D, onto M. R. C. Buck has remarked (4), that this mapping fails to be one-one, in the light of a classical theorem of Pick. If the points of βD are represented by filters of subsets of D, we can identify those filters which are sufficiently close in terms of the hyperbolic metric on D in an attempt to get a one-one correspondence between filters and points of M.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document