Rapidly growing entire functions with three singular values
We settle the problem of finding an entire function with three singular values whose Nevanlinna characteristic dominates an arbitrarily prescribed function.
math.CV↗
arXiv subjects
Sergei Merenkov
Publications and source records attributed to Sergei Merenkov.
On the Cauchy transform of weighted Bergman spaces
The problem of describing the range of a Bergman space B_2(G) under the Cauchy transform K for a Jordan domain G was solved by Napalkov (Jr) and Yulmukhametov. It turned out that K(B_2(G))=B_2^1(C\bar G) if and only if G is a quasidisk; here B_2^1(C\bar G) is the Dirichlet space of the complement of \bar G. The description of K(B_2(G)) for an integrable Jordan domain is given in [S. Merenkov, "On the Cauchy transform of the Bergman space", Mat. Fiz. Anal. Geom., 7 (2000), no. 1, 119-127]. In the present paper we give a description of K(B_2(G,ω)) analogous to the one given in [S. Merenkov, "On the Cauchy transform of the Bergman space", Mat. Fiz. Anal. Geom., 7 (2000), no. 1, 119-127] for a weighted Bergman space B_2(G,ω) with a weight ω which is constant on level lines of the Green function of G. In the case G=D, the unit disk, and under some additional conditions on the weight ω, K(B_2(D,ω))=B_2^1(C\bar{D}, ω^{-1}), a weighted analogue of a Dirichlet space.
math.CV↗
A Sierpiński carpet with the co-Hopfian property
Motivated by questions in geometric group theory we define a quasisymmetric co-Hopfian property for metric spaces and provide an example of a metric Sierpiński carpet with this property. As an application we obtain a quasi-isometrically co-Hopfian Gromov hyperbolic space with a Sierpiński carpet boundary at infinity. In addition, we give a complete description of the quasisymmetry group of the constructed Sierpiński carpet. This group is uncountable and coincides with the group of bi-Lipschitz transformations.
math.MG↗
Planar relative Schottky sets and quasisymmetric maps
A relative Schottky set in a planar domain D is a subset of D obtained by removing from D open geometric discs whose closures are in D and are pairwise disjoint. In this paper we study quasisymmetric and related maps between relative Schottky sets of measure zero. We prove, in particular, that quasisymmetric maps between such sets in Jordan domains are conformal, locally bi-Lipschitz, and that their first derivatives are locally Lipschitz. We also provide a locally bi-Lipschitz uniformization result for relative Schottky sets in Jordan domains and establish rigidity with respect to local quasisymmetric maps for relative Schottky sets in the unit disc.
math.MG↗
Hausdorff dimension of three-period orbits in Birkhoff billiards
We prove that the Hausdorff dimension of the set of three-period orbits in classical billiards is at most one. Moreover, if the set of three-period orbits has Hausdorff dimension one, then it has a tangent line at almost every point.
math.DS↗
Quasisymmetric Koebe Uniformization
We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely connected domain in the standard 2-sphere is quasisymmetrically equivalent to a circle domain if and only if X is linearly locally connected and its completion is compact. We also give a counterexample in the countably connected case.
math.MG↗
Rigidity of Schottky sets
We call a complement of a union of at least three disjoint (round) open balls in the unit sphere S^n a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a Mobius transformation on S^n. In the other direction we show that every Schottky set in S^2 of positive measure admits non-trivial quasisymmetric maps to other Schottky sets. These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries.
math.MG↗
Quasisymmetric rigidity of square Sierpinski carpets
We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpiński carpet $S_3$ is a Euclidean isometry. For carpets in a more general family, the standard $1/p$-Sierpiński carpets $S_p$, $p\ge 3$ odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that $S_p$ and $S_q$ are quasisymmetrically equivalent only if $p=q$. The main tool in the proof for these facts is a new invariant---a certain discrete modulus of a path family---that is preserved under quasisymmetric maps of carpets.
math.CV↗
A Hyperbolic Surface With A Square Grid Net
We prove the existence of a hyperbolic surface spread over the sphere for which the projection map has all its singular values on the extended real line, and such that the preimage of the extended real line under the projection map is homeomorphic to the square grid in the plane. This answers a question raised by È. B. Vinberg.
math.CV↗
Nevanlinna functions with real zeros
We show that for every non-negative integer d, there exist differential equations w''+Pw=0, where P is a polynomial of degree d, such that some non-trivial solution w has all zeros real.
math.CV↗
Maps conjugating holomorphic maps in C^n
If f is a bijection from C^n onto a complex manifold M, which conjugates every holomorphic map in C^n to an endomorphism in M, then we prove that f is necessarily biholomorphic or antibiholomorphic. This extends a result of A. Hinkkanen to higher dimensions. As a corollary, we prove that if there is an epimorphism from the semigroup of all holomorphic endomorphisms of C^n to the semigroup of holomorphic endomorphisms in M, or an epimorphism in the opposite direction for a doubly-transitive M, then it is given by conjugation by some biholomorphic or antibiholomorphic map. We show also that there are two unbounded domains in C^n with isomorphic endomorphism semigroups but which are neither biholomorphically nor antibiholomorphically equivalent.
math.CV↗
A negative answer to Nevanlinna's type question and a parabolic surface with a lot of negative curvature
Consider a simply connected Riemann surface represented by a Speiser graph. Nevanlinna asked if the type of the surface is determined by the mean excess of the graph: whether mean excess zero implies that the surface is parabolic and negative mean excess implies that the surface is hyperbolic. Teichmuller gave an example of a hyperbolic simply connected Riemann surface whose mean excess is zero, disproving the first of these implications. We give an example of a simply connected parabolic Riemann surface with negative mean excess, thus disproving the other part. We also construct an example of a complete, simply connected, parabolic surface with nowhere positive curvature such that the integral of curvature in any disk about a fixed basepoint is less than -epsilon times the area of the disk, where epsilon > 0 is some constant.
math.CV↗