On the Heesch number for the hyperbolic plane

We prove that there exists a polygon with arbitrary Heesch number on the hyperbolic plane.     

Percolation in the hyperbolic plane

We study percolation in the hyperbolic plane and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, , there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, , there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, , there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.

     

Coordinates in the hyperbolic plane

Abstract This note shows that in the hyperbolic plane three kinds of coordinates are possible. Keywords: Hyperbolic plane, Quasi-regular quadrangle, Pseudo-parallelogram, Hypercycle, Hypocycle Mathematics Subject Classification (2000): 20N05     

Analogs of the plane axioms for locally conformally Kähler manifolds

Translated from Matematicheskie Zametki, Vol. 57, No. 1, pp. 48–56, January, 1995.     

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

Support function and hyperbolic plane

Minkowskis theorem _C(cosh d(o, ) – kS) ds = 0 in the hyperbolic plane (Kleins model) for smoothly bounded horocyclic convex bodies K with outer unit normal vector u and curvature |k| 1 of C K with arclength s where S <sinh d(o, ) grad d(o, ), u> motivates the introduction of a hyperbolic support function H of K. Hereby H() d(l(), D⁺()) is the distance of the K-supporting distance curve D⁺() from the line l() through the origin o with the direction angle . – The paper deals with the representation of C, s and k by H including extremal cases and an application of Minkowskis theorem to the characterization of circles by inequalities for their hyperbolic support function.     

Polar actions on the complex hyperbolic plane

We classify the polar actions on the complex hyperbolic plane ${\mathbb{C} H^2}$ up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity 1 and four polar actions of cohomogeneity 2.     

Estimates for the hyperbolic metric of the punctured plane and applications

The hyperbolic metrich ^Ω of the twice punctured complex plane Ω is studied. A new recursive algorithm for computing the density λ_Ω ofh ^Ω is given. For a proper subdomainG of Ω we answer a question of G. Martin concerning quasiconformal mappings ofG that can be extended to the complement ofG as the identity map.     

Entropy of non-uniformly hyperbolic plane billiards

We prove exact formulas for measure theoretic entropy of plane billiards systems with absolutely-focusing boundaries with non-vanishing Lyapunov exponents. In particular, our formulas hold for the billiards introduced by Wojtkowski, Markarian, Donnay and Bunimovich. As an illustration, we calculate the entropy of a “perturbation” of the boundary of a polygon by absolutely focusing “ripples”.     

Asymptotic properties of eigenfunctions— the hyperbolic plane

Dedicated to Professor Shmuel Agmon     

Boundary value problems for a third-order hyperbolic equation on the plane

For a factorized third-order hyperbolic equation on the plane, we obtain sufficient conditions for the solvability of some boundary value problems with conditions that have not been considered for this equation earlier.     

Eigenvalue asymptotics for the Schrödinger operators on the hyperbolic plane

In this paper we consider the Schrödinger operator on the hyperbolic plane , where is the hyperbolic Laplacian and V is a scalar potential on . It is proven that, under an appropriate condition on V at ‘infinity’, the number of eigenvalues of H_V less than λ is asymptotically equal to the canonical volume of the quasi-classically allowed region as λ→∞. Our proof is based on the probabilistic methods and the standard Tauberian argument as in the proof of Theorem 10.5 in Simon (Functional Integration and Quantum Physics, Academic Press, New York, 1979).