The Hyperbolic Metric on the Complement of the Integer Lattice Points in the Plane

2017 ◽  
pp. 247-252 ◽  
Author(s):  
Katsuhiko Matsuzaki
Keyword(s):  
Integer Lattice ◽  
Hyperbolic Metric ◽  
Lattice Points ◽  
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.

scholarly journals DISTRIBUTION OF INTEGER LATTICE POINTS IN A BALL CENTRED AT A DIOPHANTINE POINT

Mathematika ◽  
2009 ◽  
Vol 56 (1) ◽  
pp. 118-134 ◽  
Author(s):  
Hyunsuk Kang ◽  
Alexander V. Sobolev
Keyword(s):  
Integer Lattice ◽  
Lattice Points

The number of non-homogeneous lattice points in subsets of Rn

1977 ◽  
Vol 82 (2) ◽  
pp. 265-268 ◽  
Author(s):  
E. S. Barnes ◽  
Michael Mather
Keyword(s):  
Integer Lattice ◽  
Lattice Points ◽  
Homogeneous Lattice

Let Zn denote the integer lattice in Rn, let A be a non-singular n × n matrix and ʗ ∈ Rn. Then G = AZn + ʗ is called a grid (non-homogeneous lattice) and its determinant det G is defined to be |det A|.

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