Matti Vuorinen
Our goal is to provide a survey of some topics in quasiconformal analysis of current interest. We try to emphasize ideas and leave proofs and technicalities aside. Several easily stated open problems are given. Most of the results are joint work with several coauthors.
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Quasiconformal homeomorphisms of the unit ball $B^n$ of ${\mathbb R}^n, n \ge 3,$ onto itself with identity boundary values are studied. A spatial analogue of Teichmüller's theorem is proved.
Here we give an alternate proof of a sufficient condition due to J. Mateu, J. Orobitg, and J. Verdera for a quasiconformal map of the plane with dilatation supported in a smooth domain to be bi-Lipschitz. We also extend this theorem to cover boundaries with certain types of corners.
We establish that every $K$-quasiconformal mapping of $w$ of the unit disk $\ID$ onto a $C^2$-Jordan domain $Ω$ is Lipschitz provided that $Δw\in L^p(\ID)$ for some $p>2$. We also prove that if in this situation $K\to 1$ with $\|Δw\|_{L^p(\ID)}\to 0$, and $Ω\to \ID$ in $C^{1,α}$-sense with $α>1/2,$ then the bound for the Lipschitz constant tends to $1$. In addition, we provide a quasiconformal analogue of the Smirnov absolute continuity result over the boundary.
Barrlund's distance function and quasiconformal maps
本文研究Barrlund度量，给出其相对于其他度量的精确界，并证明拟共形映射下该度量的精确形变结果。
Quasiconformal curves and quasiconformal maps in metric spaces
本文研究度量空间中拟共形曲线与拟共形映射的关系，证明拟共形曲线满足解析定义及模不等式下界，给出满足度量定义的充分条件，并表明拟共形映射在适当假设下是拟共形曲线，且内蕴度量下结论依然成立。
We study quasiconformal maps on non-rigid Carnot groups equipped with Carnot metric. We show that for most non-rigid Carnot groups N, all quasiconformal maps on N must be biLipschitz.
Geometric Function Theory on Uniformly Quasiconformally Homogeneous Domains
本文研究了一致拟共形齐性域上的几何函数理论，证明了拟双曲度量下拟共形性与拟对称性等价，并揭示了到 $S^n$ 或 $\mathbb{R}^n$ 的正规拟正则映射具有一致 Hölder 条件等几何性质。
Bohr-type inequalities for classes of analytic maps and K-quasiconformal harmonic mappings
本文通过结合经典Bohr不等式的改进版本与面积测度方法，推广并精化了解析自映射类及K-拟共形调和映射类的Bohr型不等式，并证明结果最优。
We show that quasiconformal maps on many Carnot groups must be biLipschitz. In particular, this is the case for 2-step Carnot groups with reducible first layer. These results have implications for the rigidity of quasiisometries between negatively curved solvable Lie groups.
It is proved the following theorem, if $w$ is a quasiconformal harmonic mappings between two Riemann surfaces with smooth boundary and aproximate analytic metric, then $w$ is a quasi-isometry with respect to Euclidean metric.
Length control for foliations by Jordan curves via smooth quasiconformal maps
针对单连通域上Jordan曲线叶状结构的长度控制问题，构造光滑5-拟共形映射，以ε=2/157肯定回答了Bouchala与Koskela提出的p=2及附近的可积性问题。
We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities.
We describe all quasiconformal maps on the higher (real and complex) model Filiform groups equipped with the Carnot metric, including non-smooth ones. These maps have very special forms. In particular, they are all biLipschitz and preserve multiple foliations. The results in this paper have implications to the large scale geometry of nilpotent Lie groups and negatively curved solvable Lie groups.
Teichmüller's classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself, and maps a given point of the domain to another given point of the domain. For a domain $D \subset {\mathbb R}^n\,,n\ge 2\,,$ we consider the class of all $K$- quasiconformal maps of $D$ onto itself with identity boundary values and Teichmüller's problem in this context. Given a map $f$ of this class and a point $x\in D\,,$ we show that the maximal dilatation of $f$ has a lowe…
Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least $2$ and that $D\subset E$ and $D'\subset E'$ are domains. In this paper, we establish, in terms of the $j_D$ metric, a necessary and sufficient condition for the homeomorphism $f: E \to E'$ to be FQC. Moreover, we give, in terms of the $j_D$ metric, a sufficient condition for the homeomorphism $f: D\to D'$ to be FQC. On the other hand, we show that this condition is not necessary.
Let E be a compact set in the plane, g be a K-quasiconformal map, and let 0<t<2. Then H^t (E) = 0 implies H^{t'} (g E) = 0, for t'=[2Kt]/[2+(K-1)t]. This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala in his celebrated paper on area distortion for quasiconformal maps and answers in the positive a Conjecture of K. Astala in op. cit.
We prove existence of solutions to a nonlinear transport equation in the plane, for which the velocity field is obtained as the convolution of the classical Cauchy Kernel with the unknown. Even though the initial datum is bounded and compactly supported, the velocity field may have unbounded divergence. The proof is based on the compactness property of quasiconformal mappings.
Stretching and Rotation Sets of Quasiconformal Mappings
本文研究平面拟共形映射的拉伸与旋转奇性，构造临界维数下非σ有限且具正Riesz容量及正规范Hausdorff测度的集合，改进已有结果。
For the result on 1-quasiconformal maps, see the paper by Cowling and Ottazzi. The result on quasiconformal maps on Carnot groups with reducible first layer will appear in a forthcoming paper by Enrico Le Donne and Xiangdong Xie.
We study one-parameter curves on the universal Teichmüller space $T$ and on the homogeneous space $M=\Diff S^1/\Rot S^1$ embedded into $T$. As a result, we deduce evolution equations for conformal maps that admit quasiconformal extensions and, in particular, such that the associated quasidisks are bounded by smooth Jordan curves. Some applications to Hele-Shaw flows of viscous fluids are given.
The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics.
In this article, we obtain quasiconformal extensions of some classes of conformal maps defined either on the unit disc or on the exterior of it onto the extended complex plane. Some of these extensions have been obtained by constructing suitable Loewner chains and others have been obtained by applying a well-known result.
R. Fehlmann and M. Vuorinen proved in 1988 that Mori's constant $M(n,K)$ for $K$-quasiconformal maps of the unit ball in $\mathbf{R}^n$ onto itself keeping the origin fixed satisfies $M(n,K) \to 1$ when $K\to 1 .$ We give here an alternative proof of this fact, with a quantitative upper bound for the constant in terms of elementary functions. Our proof is based on a refinement of a method due to G.D. Anderson and M. K. Vamanamurthy. We also give an explicit version of the Schwarz lemma for quasiconformal self-maps of the unit disk. Some experimental results are provided to compare the various …
Piecewise quasiconformal dynamical systems of the unit circle
本文研究单位圆上分段拟共形覆盖映射，给出共轭可拟共形或David扩张到单位圆盘的充分条件，推广了分段解析情形，并应用于映射分类、Blaschke乘积共形配对及抛物 basins 几何。
We study the mean radius growth function for quasiconformal mappings. We give a new sub-class of quasiconformal mappings in $\mathbb{R}^n$, for $n\geq 2$, called bounded integrable parameterization mappings, or BIP maps for short. These have the property that the restriction of the Zorich transform to each slice has uniformly bounded derivative in $L^{n/(n-1)}$. For BIP maps, the logarithmic transform of the mean radius function is bi-Lipschitz. We then apply our result to BIP maps with simple infinitesimal spaces to show that the asymptotic representation is indeed quasiconformal by showing t…
Indecomposable Quasiconformal Maps of Manifolds
本文研究了闭合manifold上不可分解的准共形映射的存在性，核心方法是构造具有任意小共形扭曲的映射组合，主要贡献是证明了此类映射的不可分解性。
Existence of quasiconformal maps with maximal stretching on any given countable set
本文针对 $\mathbb{R}^d$ 中任意有界可数集构造在每一点均达到最大拉伸的 $K$-拟共形映射，从而将平面极值构造推广至一般维数。
We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala's conjecture that the Hausdorff dimension of a $k$-quasicircle is at most $1+k^2$.
We establish sharp bounds for simultaneous local rotation and Hölder-distortion of planar quasiconformal maps. In addition, we give sharp estimates for the corresponding joint quasiconformal multifractal spectrum, based on new estimates for Burkholder functionals with complex parameters. As a consequence, we obtain optimal rotation estimates also for bi-Lipschitz maps.
An important open problem in geometric complex analysis is to find algorithms for explicit determination of basic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmuller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of such polygons with the geometry of the universal Teichmuller space. We provide a new approach, based on affine transformations of univalent functions.
Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2 and that $D\varsubsetneq E$ and $D'\varsubsetneq E'$ are uniform domains with homogeneously dense boundaries. We consider the class of all $φ$-FQC (freely $φ$-quasiconformal) maps of $D$ onto $D'$ with bilipschitz boundary values. We show that the maps of this class are $η$-quasisymmetric. As an application, we show that if $D$ is bounded, then maps of this class satisfy a two sided Hölder condition. Moreover, replacing the class $φ$-FQC by the smaller class of $M$-QH maps, we show that $M$-QH maps with bilipschitz b…
Improved Hölder Continuity of Quasiconformal Maps
本文结合等周不等式与拟圆长度分析，改进复平面拟共形映射的 Hölder 连续性下界，刻画极值子并应用于椭圆偏微分方程。
A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$ of ${\Bbb R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:{\Bbb R}^N\to{\Bbb R}^N$ . We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz $n$-manifold and for a quasiplane to have big pieces of bi-Lipschitz images of ${\Bbb R}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension $N-n$. To establish the big pieces criterion, we prove new extension theorems for "almost affine" maps, which ar…
Hardy spaces of harmonic quasiconformal mappings and Baernstein's theorem
本文研究单位圆盘内规范化 $K$-拟共形调和映射，证明其几何子类解析与余解析部分的 Baernstein 型极值定理，并据此给出积分均值估计及属于 Hardy 空间 $h^p$ 的 $p$ 范围。
An important problem in applications of quasiconformal analysis and in its numerical aspect is to establish algorithms for explicit or approximate determination of the basic quasiinvariant curvelinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmuller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficients of associated quasicircles. We prove a general theorem of new type answering this question for univalent functions in arbitrary quasiconformal domains and provide its applications. The results are strengthen…
A counterexample to the Clunie--Sheil-Small coefficient-difference conjecture via quasiconformal maps
通过构造调和K-拟共形映射，否证了Clunie--Sheil-Small系数差猜想，并揭示了其Hardy空间性质与共形情形的差异，同时给出了Beltrami系数在Sobolev空间时的精确范围。
Extremality of hyperbolic spiral and stretch maps
该研究针对双曲平面中的双曲螺线与拉伸映射，证明了其拟共形映射的极值结果
We define Hardy spaces $H^p$, $0<p<\infty$, for quasiconformal mappings on the Korányi unit ball $B$ in the first Heisenberg group $\mathbb{H}^1$. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Korányi and Reimann, and Balogh and Tyson. First, we prove the existence of $p_0(K)>0$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential …
We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circumcircle preserving piecewise projective interpolation between discretely conformally equivalent triangulations. We show that another interpolation scheme, angle bisector preserving piecewise projective interpolation, is in a sense optimal with respect to dilatation. These two in…
We describe briefly a new approach to some problems related to Teichmüller spaces, invariant metrics, and extremal quasiconformal maps. This approach is based on the properties of plurisubharmonic functions, especially of the plurisubharmonic Green function. The main theorem gives an explicit representation of the Green function for Teichmüller spaces by the Kobayashi-Teichmüller metric of these spaces. This leads to various applications. In particular, this gives a new characterization of extremal quasiconformal maps.
The Andreev-Thurston Circle Packing Theorem is generalized to packings of convex bodies in planar simply connected domains. This turns out to be a useful tool for constructing conformal and quasiconformal mappings with interesting geometric properties. We attempt to illustrate this with a few results about uniformizations of finitely connected planar domains. For example, the following variation of a theorem by Courant, Manel and Shiffman is proved and generalized. If $G$ is an $n+1$-connected bounded planar domain, $H$ is a simply connected bounded planar domain, and $P_1,P_2,...,P_n$ are (co…
In this paper we prove that if $ϕ:\C\to\C$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \C$ is a compact set contained in a ball $B$, then $$\frac{\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\diam(B)^{\frac2{K+1}}} \geq c^{-1} (\frac{γ(ϕ(E))}{\diam(ϕ(B))})^{\frac{2K}{K+1}},$$ where $γ$ stands for the analytic capacity and $\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$ is a capacity associated to a non linear Riesz potential. As a consequence, if $E$ is not $K$-removable (i.e. removable for bounded $K$-quasiregular maps), it has positive capacity $\dot C_{frac{2K}{2K+1},\frac{2K+1}{K+1…
By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$. In this case, one wants to find the ``best" mapping between these polygons, i.e., one that minimizes the maximum angle distortion (the dilatation) over \textit{all} points in $P$. From complex analysis such maps are known to exist and are unique. They are called extremal quasiconformal maps, or Teichmüller maps. Although there are many efficient ways to comp…
Let $Γ$ be a closed Jordan curve, and $f$ the conformal mapping that sends the unit disc $\mathbb{D}$ onto the interior domain of $Γ$. If $\log f'$ belongs to the Dirichlet space $\mathcal{D}$, we call $Γ$ a Weil-Petersson curve. The purpose of this note is to extend recent results, obtained by G. Cui and Ch. Bishop in the case of Weil-Petersson curves, to the case when $\log f'$ belongs to either some $Q_{p,0},$ space, for $0<p\leq 1$, or to some weighted-Dirichlet space contained in $\mathcal{D}$. More precisely, we will characterize the quasiconformal extensions of $f$, and describe some of…
A Lewy-type theorem for pluriharmonic mappings in $\C^2$ and bi-Lipschitz quasiconformal harmonic maps
本研究证明了$\mathbb{C}^2$中多重调和一一映射的实雅可比行列式非零的四维Lewy型定理，还给出了调和拟共形同胚为双Lipschitz映射的判定准则。
Unique extremality of affine maps on plane domains
本研究证明仿射映射是复平面上分布良好集合补集上的唯一极值拟共形映射，解决了相关猜想，并通过Bergman投影与亚纯单位分解构造了所需的Reich序列。
In this article we study fine regularity properties for mappings of finite distortion. Our main theorems yield strongly localized regularity results in the borderline case in the class of maps of exponentially integrable distortion. Analogues of such results were known earlier in the case of quasiconformal mappings. Moreover, we study regularity for maps whose distortion has higher exponential integrability.
We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let $(F_j)_{j \geq 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\overline{\partial} F_j (z)=μ_j(z,ω) \partial F_j(z),$ where the random dilatation satisfies $|μ_j|\leq k<1$ and has locally periodic statistics, for example of the type $$μ_j (z,ω)=ϕ(z)\sum_{n\in \mathbf{Z}^2}g(2^j z-n,X_{n}(ω)), $$ where $g(z,ω)$ decays rapidly in $z$, the random variables $X_{n}$ are i.i.d…
We present a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author's interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the …
We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $μ$ on such a space, we introduce $μ$-quasiconformal maps $f:X \to \mathbb R^2$, whose definition involves deforming lengths of curves by $μ$. We show that if $μ$ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $μ$-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinite…
The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. The underlying geometric category is based on the moduli space of Riemann surfaces with parametrized boundary components and the sewing operation. We survey the recent and careful study of these objects, which has led to significant connections with quasiconformal Teichmuller theory and geometric function theory. In particular we propose that the natural analytic setting for conformal field theory is the moduli space of Riemann surfaces with so-called Weil-Petersson class parametri…
David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show \vs {enumerate} [$\bullet$] Given $α$ and $β$ in $[0,2]$, there exists a David map $ϕ:\CC \to \CC$ and a compact set $Λ$ such that $\Hdim Λ=α$ and $\Hdim ϕ(Λ)=β$. \vs [$\bullet$] There exists a David map $ϕ:\CC \to \CC$ such that the Jordan curve $Γ=ϕ(\Sen)$ satisfies $\Hdim Γ=2$.\vs {enumerate} One should contrast the first statement with the fact that quasiconformal m…
A geometric property of quadrilaterals
针对平面拟共形映射中的四边形，本文证明模处于 $[1/K,K]$ 的四边形内部必含一个半径由对边内部距离和 $K$ 决定的圆盘。
Bijective Density-Equalizing Quasiconformal Map for Multiply-Connected Open Surfaces
本文提出一种基于拟共形流的双射密度均衡展平映射方法，通过优化Beltrami系数和迭代调整目标圆域，解决多连通开曲面的密度均衡参数化问题，并支持地标约束，在图形学和医学成像中验证有效。
Free-Boundary Quasiconformal Maps via a Least-squares Operator in Diffeomorphism Optimization
本文提出基于最小二乘准共形算子的自由边界微分同胚优化方法，通过谱Beltrami网络实现高效优化，提升几何建模与生物成像的精度与稳定性。
Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichmüller space; John Milnor, Thurston's algorithm without critical finiteness; Mary Rees, A possible approach to a complex renormalization problem. 2. Geometry of Julia Sets: Lennart Carleson, Geometry of Julia sets; John Milnor, Problems on local connectivity. 3. Measurable Dynamics: Mikhail Lyubich, Measure and Dimension of Julia Sets; Feliks Przytycki, On invariant measures for iterations of holomorphic maps. 4. Iterates of Entire Functions: Robert De…
Extremal length and duality
本文集中于极值长度理论的对偶性性质及其在拟共形分析中的应用，展示了该理论的灵活性，并给出了度量曲面一致化、$n$ 维空间间的拟共形映射及多重连通平面区域间共形映射三种背景下的最新进展。
A variational inequality for the images of $k$-dimensional hyperplanes under quasiconformal maps of the $n$-dimensional Euclidean space is proved when $1\le k\le n-2 .$
Let $f$ be the infinitesimal generator of a one-parameter semigroup $\left\{ F_{t}\right\} _{t\ge0}$ of holomorphic self-mappings of the open unit disk $Δ$. In this paper we study properties of the family $R$ of resolvents $(I+rf)^{-1}:Δ\toΔ~ (r\ge0)$ in the spirit of geometric function theory. We discovered, in particular, that $R$ forms an inverse Löwner chain of hyperbolically convex functions. Moreover, each element of $R$ satisfies the Noshiro-Warschawski condition and is a starlike function of order at least $\frac12$,. This, in turn, implies that each element of $R$ is also a holomorphi…
Markov maps and quasi-conformal deformations associated to geometrically finite Kleinian groups
本文研究带抛物元素的几何有限Kleinian群的拟共形形变，证明其极限集上的首次返回映射具扩张性且该性质在拟共形形变下不变，应用其证明该类群具Ruelle性质，还通过特征值算法计算Hecke群Hausdorff维数以支撑结果。
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