Ilia Binder, Mark Braverman, Michael Yampolsky
In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of boundary is given either as a list of pixels, or by a Turing Machine.
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This article presents a clear proof of the Riemann Mapping Theorem via Riemann's method, uncompromised by any appeals to topological intuition.
In this paper we give a new proof of Riemann's well known mapping theorem. The suggested method permits to prove an analog of that theorem for the three dimensional case.
It is proved criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.
In the present paper, we investigate non-homeomorphic mappings of Riemannian surfaces of Sobolev class. We have obtained some estimates of distortion of moduli of families of curves. As consequence, we have obtained results about the boundary behavior of such mappings between domains on Riemannian surfaces.
We completely describe the asymptotic behaviour of the Riemann mapping function and its derivatives at an analytic cusp. We achieve the same for the inverse of the mapping function.
We prove a Schwarz type lemma for harmonic mappings between the unit and a geodesic line in a Riemenn surface.
In terms of dilatations, it is proved a series of criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between regular domains on the Riemann surfaces
We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds with universal cover a bounded domain is a finite set.
We study some problems related to the boundary behavior of maps of domains of Riemannian surfaces. In particular, for mappings satisfying the inverse Poletsky type modulus inequality, we establish the possibility of their continuous extension to the boundary in terms of prime ends. We also study the local behavior of such mappings at boundary points.
We investigate non-homeomorphic mappings of Riemannian surfaces of Sobolev class. We have obtained some estimates of distortion of moduli of families of curves. We have proved that, under some conditions, these mappings have a continuous extension to a boundary of a domain in terms of prime ends.
A boundary behavior of ring mappings on Riemannian manifolds, which are generalization of quasiconformal mappings by Gehring, is investigated. In terms of prime ends, there are obtained theorems about continuous extension to a boundary of classes mentioned above. In the terms mentioned above, there are obtained results about equicontinuity of these classes in the closure of the domain.
Theorems on continuous extension on boundary for one class of open discrete mappings between Riemannian manifolds are obtained. In particular, there is proved that, open discrete ring $Q$-mappings $f:D\rightarrow D^{\,\prime}$ are extend to $\partial D$ whenever $\partial D$ is locally connected, $\partial D^{\,\prime}$ is strongly accessible, and a function $Q$ has finite mean oscillation at $\partial D.$
We consider mappings of domains of Riemannian manifolds that admit branch points and satisfy a certain condition regarding the distortion of the modulus of families of paths. We have established logarithmic estimates of distance distortion under such mappings. A separate study relates to the situation when the mappings are defined in metric spaces, and one of them is the Loewner space. We also studied the question of equicontinuity of the families of the indicated mappings in the closure of the domain. In addition, we have established the possibility of continuous extension of these mappings t…
For some class of mappings, there are investigated problems connected with a possibility of continuous extension to a boundary on Riemannian manifolds. In particular, for so-called ring mappings, there is proved a result related to continuous extension to an isolated boundary point. Besides that, similar theorems hold for more general boundaries. As application of developed technique, there is proves a theorem about continuous extension of mapping of Orlicz--Sobolev class to an isolated singularity.
In this paper we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz-Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and billiard motions in a regular stellated $n$-gon in the complex plane.
The primary purpose of the paper is to study how a Riemann mapping depends on the corresponding Jordan curve. We are mainly concerned with those Jordan curves in the Weil-Petersson class, namely, the corresponding Riemann mappings can be quasiconformally extended to the whole plane with Beltrami coefficients being square integrable under the Poincaré metric. We endow the space of all normalized Weil-Petersson curves with a new real Hilbert manifold structure and show that it is topologically equivalent to the standard complex Hilbert manifold structure.
We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: it is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping in $H^1$, even for non-Delaunay triangulations. Additionally, for Delaunay triangulations the discrete conformal maps converge …
In our previous work, for a given Riemann surface $Y_0$ with marked handle, we investigated geometric properties of the set of marked once-holed tori $X$ allowing holomorphic mappings of $X$ into $Y_0$. It turned out that it is a closed domain with Lipschitz boundary. In the present paper we show that the boundary is never smooth. Also, we evaluate the critical extremal length for the existence of holomorphic mappings in terms of hyperbolic lengths.
In this article we investigate the century-old continuous extension problem of the Riemann map. Let $G$ be a simply connected domain. We call $λ$ in $\partial G$ a multiple point if there are simply connected subdomains $ U$ and $V$ such that $λ\in\partial U \cap\partial V$ and $ dist (\partial U\cap G , \partial V\cap G )>0$. We show that the Riemann map of $G$ has a continuous extension to $\overline G$ if and only if $\partial G$has no multiple points. All of the results in this paper, together with the Riemann mapping theorem, give a complete and desirable solution to the mapping problem t…
We show that on bounded Lipschitz pseudoconvex domains that admit good weight functions the $\overline{\partial}$-Neumann operators $N_q, \overline{\partial}^* N_{q}$, and $\overline{\partial} N_{q}$ are bounded on $L^p$ spaces for some values of $p$ greater than 2.
We discuss applications of an improvement on the Riemann mapping theorem which replaces the unit disc by another "double quadrature domain," i.e., a domain that is a quadrature domain with respect to both area and boundary arc length measure. Unlike the classic Riemann Mapping Theorem, the improved theorem allows the original domain to be finitely connected, and if the original domain has nice boundary, the biholomorphic map can be taken to be close to the identity, and consequently, the double quadrature domain close to the original domain. We explore some of the parallels between this new th…
We consider the space of chord-arc curves on the plane passing through the infinity with their parametrization $γ$ on the real line, and embed this space into the product of the BMO Teichmüller spaces. The fundamental theorem we prove about this representation is that $\log γ'$ also gives a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Rie…
We describe the asymptotic behaviour and the dependence on the regularization of logarithmically divergent integrals of products of meromorphic and antimeromorphic forms on complex manifolds. Our formula is expressed in terms of residues of Dolbeault forms, a notion introduced in this paper. The proof is based on a result on the asymptotic behaviour of Riemann mappings of small domains.
Recall that the Hilbert (Riemann-Hilbert) boundary value problem for the Beltrami equations was recently solved for general settings in terms of nontangential limits and principal asymptotic values. Here it is developed a new approach making possible to obtain new results on tangential limits in multiply connected domains. It is shown that the spaces of the found solutions have the infinite dimension for prescribed families of Jordan arcs terminating in almost every boundary point. We give also applications of results obtained by us for the Beltrami equations to the boundary value problems of …
The subject of this paper is Beurling's celebrated extension of the Riemann mapping theorem \cite{Beu53}. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem contains a number of gaps which seem inherent in Beurling's geometric and approximative approach. We provide a complete proof of the Beurling-Riemann mapping theorem by combining Beurling's geometric method with a number of new analytic tools, notably $H^p$-space techniques and methods from the theory of Riemann-Hilbert-Poincaré problems. One additional advantage of this approach is …
We show how to express a conformal map of a general two connected domain in the plane such that neither boundary component is a point to a representative domain which has the virtue of having an explicit algebraic Bergman kernel function. We shall explain why the representative domain is the best analogue of the unit disc in the two connected setting. The conformal map will be given as a simple and explicit algebraic function of an Ahlfors map of the domain associated to a specially chosen point. It will follow that the conformal map can be found by solving the same extremal problem that deter…
Let S be a generic C-infinity smooth CR manifold in C^n, n > 1, and let M be a generic C-infinity CR submanifold of S X C^m. We prescribe conditions on M so that it is the disjoint union of graphs of CR maps f:S-->C^m. We also consider the special case where S is the boundary of a bounded, smoothly bounded open set D in C^n, n > 1. In this case we obtain conditions which guarantee that such an f above extends to be analytic on D. This provides a solution to a particular nonlinear Riemann-Hilbert boundary value problem for analytic functions.
We consider a Jordan arc Γin the complex plane \mathbb{C} and a regular measure μwhose support is Γ. We denote by D the upper Hessenberg matrix of the multiplication by z operator with respect to the orthonormal polynomial basis associated with μ. We show in this work that, if the Hessenberg matrix D is uniformly asymptotically Toeplitz, then the symbol of the limit operator is the restriction to the unit circle of the Riemann mapping function ϕ(z) which maps conformally the exterior of the unit disk onto the exterior of the support of the measure μ. We use this result to show how to approxima…
Let $ρ_Σ=h(|z|^2)$ be a metric in a Riemann surface $Σ$, where $h$ is a positive real function. Let $\mathcal H_{r_1}=\{w=f(z)\}$ be the family of univalent $ρ_Σ$ harmonic mapping of the Euclidean annulus $A(r_1,1):=\{z:r_1< |z| <1\}$ onto a proper annulus $A_Σ$ of the Riemann surface $Σ$, which is subject of some geometric restrictions. It is shown that if $A_Σ$ is fixed, then $\sup\{r_1: \mathcal H_{r_1}\neq \emptyset \}<1$. This generalizes the similar results from Euclidean case. The cases of Riemann and of hyperbolic harmonic mappings are treated in detail. Using the fact that the Gauss m…
We prove in two different ways that the monodromy map from the space of irreducible $\mathfrak{sl}_2$-differential-systems on genus two Riemann surfaces, towards the character variety of $\mathrm{SL}_2$-representations of the fundamental group, is a local diffeomorphism. This is motivated by a question raised by Étienne Ghys about Margulis' problem: existence of curves of negative Euler characteristic in compact quotients of $\mathrm{SL}_2(\mathbb{C})$.
Global mapping properties of the Riemann Zeta function are used to investigate its non trivial zeros.
We show that the structure of proper holomorphic maps between the $n$-fold symmetric products, $n\geq 2$, of a pair of non-compact Riemann surfaces $X$ and $Y$, provided these are reasonably nice, is very rigid. Specifically, any such map is determined by a proper holomorphic map of $X$ onto $Y$. This extends existing results concerning bounded planar domains, and is a non-compact analogue of a phenomenon observed in symmetric products of compact Riemann surfaces. Along the way, we also provide a condition for the complete hyperbolicity of all $n$-fold symmetric products of a non-compact Riema…
We study the boundary behavior of finitely Bi-Lipschitz mappings on smooth Riemannian manifolds
Let Σbe a compact Riemann surface with n distinguished points p_1,...,p_n. We prove that the set of n-tuples (ϕ_1,...,ϕ_n) of univalent mappings ϕ_i from the open unit disc into Σmapping 0 to p_i, with non-overlapping images and quasiconformal extensions to a neighbourhood of the closed unit disk, carries a natural complex Banach manifold structure. This complex structure is locally modelled on the n-fold product of a two complex-dimensional extension of the universal Teichmueller space. Our results are motivated by Teichmueller theory and two-dimensional conformal field theory.
Removable singularities of mappings with inverse Poletksy inequality on Rie\-man\-nian manifolds
本文研究Riemannian流形上满足逆Poletsky型模不等式的开离散映射，证明在遗漏至少两点且控制函数几乎处处球面可积时，边界孤立点为可去奇点。
Let $M$ and $N$ be doubly connected Riemann surfaces with $\mathscr{C}^{1,α}$ boundaries and with nonvanishing conformal metrics $σ$ and $\wp$ respectively, and assume that $\wp$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. Assume that ${\Ho}_ρ(M, N)$ is the class of all $\mathscr{W}^{1,2}$ bomeomorphisms between $M$ and $N$ and assume that $\mathcal{E}^\wp: \overline{\Ho}_ρ(M, N)\to \mathbf{R}$ is the Dirichlet-energy functional, where $\overline{\Ho}_ρ(M, N)$ is the closure of ${\Ho}_ρ(M, N)$ in $\mathscr{W}^{1,2}(M,N)$. By using a result of Iwaniec, Kovalev…
The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this explanation, the hyperbolicity of renormalization, has not been proved yet. The present work considers a cylinder renormalization - a novel type of renormalization for holomorphic maps with a Siegel disk w…
The study of the global mapping properties of arbitrary Dirichlet L-functions is undertaken. The results are applied to the proof of the Generalized Riemann Hypothesis.
Efficient Numerical Conformal Mappings on Multiply Connected Riemann Surfaces
本文推广并改进了共轭函数法，用于多连通平面区域和黎曼曲面上的数值共形映射计算，通过高阶有限元方法实现了高精度，并推导了互逆误差估计。
We give an algorithm for finding conformal mappings onto the upper half-plane and conformal modules of some types of polygons. The polygons are obtained by stretching along the real axis polyominoes i.e., polygons which are connected unions of unit squares with vertices from the integer lattice. We consider the polyominoes of two types, so-called the $P$-pentomino and the $L$-tetromino. The proofs are based on the Riemann-Schwarz reflection principle and uniformization of compact simply-connected Riemann surfaces by rational functions.
Conformal Riemann mapping of the unit disk onto a simply-connected domain $W$ is a central object of study in classical Complex Analysis. The first complete proof of the Riemann Mapping Theorem given by P. Koebe in 1912 is constructive, and theoretical aspects of computing the Riemann map have been extensively studied since. Carath{é}odory Theory describes the boundary extension of the Riemann map. In this paper we develop its constructive version with explicit complexity bounds.
We prove a result on the structure of finite proper holomorphic mappings between complex manifolds that are products of hyperbolic Riemann surfaces. While an important special case of our result follows from the ideas developed by Remmert and Stein, the proof of the full result relies on the interplay of the latter ideas and a finiteness theorem for Riemann surfaces.
On the properness of $p$-conformal energy on the Teichmüller space of a Riemann surface
该研究证明了p-共形能量在固定亏格黎曼曲面的泰希米勒空间上为正当泛函，其核心依赖于对双曲环面间有限畸变唯一极值映射及其p-共形能量的显式识别结果。
Scattering theory on Riemann surfaces II: The scattering matrix and generalized period mappings
本文基于曲线系统边值问题与跳跃问题构建黎曼曲面调和一形式散射理论，给出由 Schiffer 算子表示的酉散射矩阵，并通过正极化 Lagrangian 空间统一经典极化与万有 Teichmüller 空间周期映射。
We propose an analytical approach to the conformal mapping of (rectangular) polygons based on the theory of Riemann surfaces and theta functions.
Similarity surfaces, connections, and the measurable Riemann mapping theorem
本文通过构造由多边形粘合的相似曲面序列，研究复平面上Beltrami方程解的逼近，独立证明了解关于Beltrami形式的解析依赖性，并证明紧支集C²数据下Christoffel符号定义的平行移动序列收敛于特定仿射联络对应的平行移动。
The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting.
We investigate Benford's law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained as the limit of iterative sets, and the unique measures of their components follow a geometric distribution, which is Benford in most bases. Building on this intuition, we aim to study this distribution in more complicated fractals. We examine the Laurent coefficients of a Riemann mapping and the Taylor coefficients of its reciprocal function from the exterior of the Mandelbrot set to the complement of the unit disk. These coefficients are 2-adic rational nu…
We make several new contributions to the study of proper holomorphic mappings between balls. Our results include a degree estimate for rational proper maps, a new gap phenomenon for convex families of arbitrary proper maps, and an interesting result about inverse images.
Holomorphic mappings of finitely connected planar domains
本文探究无需Koebe定理证明有限连通平面区域全纯映射相关结果，给出连通度>2的有限连通区域间真全纯映射的刚性定理，还提供不依赖欧拉示性数的平面Riemann-Hurwitz公式新初等证明。
$L^p$-Extremal Teichmüller mappings between Riemann surfaces are diffeomorphisms
研究解析有限黎曼曲面间同胚类中 \(L^p\) 共形能量最小化问题，证明 \(1\leq p<\infty\) 时最小化映射存在、唯一且为微分同胚，\(p = \infty\) 时为准共形但非微分同胚。
This paper presents a new uniquely solvable boundary integral equation for computing the conformal mapping, its derivative and its inverse from bounded multiply connected regions onto the five classical canonical slit regions. The integral equation is derived by reformulating the conformal mapping as an adjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, we derive a boundary integral equation with the adjoint generalized Neumann kernel for the derivative of the boundary correspondence function $θ'$. Only the right-hand side of the integral equation is different from a ca…
In this paper we present a theoretical background of a coupled analytical-numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical-numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of the complex function theory and couple it continuously with the finite element solution in the region far from singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated …
Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation and complicated explicit computations. We also show that the so-called gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the so…
We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a ${\rm CAT}(0)$ group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus $\ge 2$) with injecti…
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