Emil Saucan
We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{n}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (torsion) elements are uniformly bounded. This is accomplished by developing a technique for mashing distinct fat triangulations while preserving fatness.
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We give a new proof of the existence of nontrivial quasimeromorphic mappings on a smooth Riemannian manifold, using solely the intrinsic geometry of the manifold.
We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.
The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.
We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{3}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (i.e torsion) elements are uniformly bounded. This is accomplished by developing a technique for meshing distinct fat triangulations while preserving fatness. We further show how to adapt the proof to higher dimensions.
We discuss the value distribution of quasimeromorphic mappings in ${\mathbb R}^n$ with given behavior in a neighborhood of an essential singularity.
While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain $Ω^*$ there exists $ε>0$ such that for every doubly connected domain $Ω$ with $\operatorname{Mod} Ω^*<\operatorname{Mod}Ω<\operatorname{Mod} Ω^*+ε$ there exists a harmonic diffeomorphism from $Ω$ onto $Ω^*$.
A proper or singular abelian mapping from $C^n$ to $\bar\C^n$ is parametrized by $n$ meromorphic functions with at most $2n$ periods. We develop the existence and structure theorems of the classical theory of an abelian mapping purely on the basis of its defining functional equation, the so-called algebraic addition theorem (AAT), with no appeal to any representation as quotients of theta functions. We offer two new proofs of the periodicity of a nonrational mapping admitting an AAT. We also prove by new arguments the existence of a rational group law on an associated algebraic variety, and th…
We give a necessary and sufficient condition for the existence of nondegenerate holomorphic mappings between pseudoellipsoidal real hypersurfaces, and provide an explicit parametrization for the collection of all such mappings (in the situations where they exist).
We present a result on existence of some kind of peak functions for $\C$-convex domains and for the symmetrized polydisc. Then we apply the latter result to show the equivariance of the set of peak points for $A(D)$ under proper holomorphic mappings. Additionally, we present a description of the set of peak points in the class of bounded pseudoconvex Reinhardt domains.
In this paper, we discuss some properties on hyperbolic-harmonic mappings in the unit ball of $\mathbb{C}^{n}$. First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz-Pick type theorem for hyperbolic-harmonic mappings and apply it to prove the existence of Landau-Bloch constant for mappings in $α$-Bloch spaces.
For an arbitrary convex function $Ψ:[1,\infty) \to [1,\infty)$, we consider uniqueness in the following two related extremal problems: Problem A boundary value problem: Establish the existence of, and describe the mapping $f$, achieving \[ \inf_f \Big\{ \int_{\Bbb D} Ψ({\Bbb K}(z,f))\; dz : f:\bar{\Bbb D} \to \bar{\Bbb D} \; \mbox{a homeomorphism in $W^{1,1}_{0}({\Bbb D})+f_0$} \Big\}. \] Here the data $f_0:\bar{\Bbb D} \to \bar{\Bbb D}$ is a homeomorphism of finite distortion with $\int_{\Bbb D} Ψ({\Bbb K}(z,f_0))\; dz<\infty$ -- a barrier. Next, given two homeomorphic Riemann surfaces $R$ an…
In this paper we establish results on the existence of nontangential limits for weighted $\Cal A$-harmonic functions in the weighted Sobolev space $W_w^{1,q}(\Bbb B^n)$, for some $q>1$ and $w$ in the Muckenhoupt $A_q$ class, where $\Bbb B^n$ is the unit ball in $\Bbb R^n$. These results generalize the ones in section \S3 of [KMV], where the weight was identically equal to one. Weighted $\Cal A$-harmonic functions are weak solutions of the partial differential equation $$\text{div}(\Cal A(x,\nabla u))=0,$$ where $αw(x) |ξ|^{q} \le < \Cal A(x,ξ),ξ>\le βw(x) |ξ|^{q}$ for some fixed $q\in (1,\inft…
We consider problems concerning the existence of solutions of the Beltrami equations and their convergence in the entire complex plane. We are mainly interested in the case when these solutions satisfy the so-called hydrodynamic normalization condition in the neighborhood of infinity. Under conditions related to dilations of inverse mappings, we have established the existence of such solutions in the class of continuous Sobolev mappings. We have also obtained results on the locally uniform limit of a sequence of such solutions.
Minimal degree rational open up mappings and related questions
本文证明了用于展开 $n$ 条弧的极小次数 $n+1$ 有理共形映射的存在性与唯一性，并探讨了具有指定临界值和临界点的极小次数有理函数的存在性及解的数量界限。
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