万有Teichmüller空间与Loewner链

通过采用构造Loewner链的方法,得到了单叶函数的充分条件,同时,结合万有Teichmüller空间理论,利用Loewner链构造了单叶函数的拟共形扩张表达式,并且得到了一些拟圆区域的单叶性内径的下界估计不等式.     

万有Teichmller空间与Loewner链

提要通过采用构造Loewner链的方法,得到了单叶函数的充分条件,同时,结合万有Teichm(u|¨)ller空间理论,利用Loewner链构造了单叶函数的拟共形扩张表达式,并且得到了一些拟圆区域的单叶性内径的下界估计不等式.     

近星映照的齐次展开式的二次项估计

该文在C~n中的单位球B_n上,首先给出了正规化星形映照的齐次展开式的二次项系数上界的较精确的估计;其次利用Loewner链的性质,给出了近星映照的齐次展开式的二次项估计.     

Roper-Suffridge延拓算子与Loewner链

主要讨论一类推广的Roper-Suffridge算子在一定条件下能够嵌入Loewner链,并从α次殆β型螺形映照的解析特征出发证明推广的Roper-Suffridge算子在一类有界完全Reinhardt域上保持α次殆β型螺形性.所得结果推广了已有的结论.     

α次殆β型螺形映照齐次展开式的精细估计

从Loewner链的角度讨论C~n中单位球B~n上α次殆β型螺形映照齐次展开式的相关项的上界,并作为特殊情况得出β型螺形映照、星形映照和α次殆星形映照齐次展开式的相关项的上界估计,推广了螺形映照及星形映照齐次展开式的二次项系数的上界估计.     

随机Loewner演化介绍

本文对随机Loewner演化(stochastic Loewner evolution, SLE)这一新的研究方向做一个综述性的介绍.随机Loewner演化是Oded Schramm于2000年前后创立的曲线上的单参数共形不变测度族理论.它与复分析、共形几何、分形几何和随机分析有非常紧密的联系,特别是在统计物理中有十分重要的应用.     

Stieltjes函数生成的广义Loewner矩阵的秩不变性

研究由Stieltjes函数生成的两类广义Loewner矩阵的秩不变性,证明了由同一Stieltjes函数生成的第一类同型的广义Loewner矩阵的秩是相等的,而生成的第二类同型的广义Loewner矩阵的秩相等或者相差1.     

推广的Roper-Suffridge算子与Loewner链

将Roper-Suffridge箅子在C~n中单位球B~n上做了进一步推广,并考察推广后的算子何时能保持双全纯映照子族的性质.利用k阶零点及双全纯映照子族的增长定理,重点研究了推广后的算子在B~n上保持α次β型螺形映照及强β型螺形映照的性质,并由调和函数的最小值原理及具有正实部函数的性质,揭示了推广后的算子能够嵌入Loewner链,从而得到推广后的算子在B~n上保持α次殆β型螺形映照的性质.     

Loewner空间上的拟共形映照

本文把Rn空间上拟共形映照的距离、模、分析定义推广到Loewner空间上,并证明了它们的等价性.     

单位球Bn上复数λ阶的g-星形映射

本文引入单位球Bⁿ?Cⁿ上复数λ阶的g-星形映射族,统一了复数λ阶的殆星映射和g-星形映射.应用Loewner链方法,建立了该映射族的增长定理,给出了k次齐次多项式P_k的数量特征,证明了Bⁿ上扰动的Roper-Suffridge算子Φ_Pk(f)(z)=(f(z₁)+P_k(z₀)f’(z₁),[f’(z₁)]^1/kz₀)^T保持复数入阶的g-星形性.本文结果不仅推广了Bⁿ上不同星形映射子族的增长定理,而且给出了扰动项P_k更为简洁的几何特征.     

Rescaled Lévy-Loewner hulls and random growth

We consider radial Loewner evolution driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov HL(0) model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1. Using a different type of compound Poisson process, where the Poisson kernel is replaced by the heat kernel, as driving function, we recover one case of the aforementioned model and SLE(κ) as limits.     

Multiple SLE and the complex Burgers equation

In this note we ask whether one can take the limit of multiple SLE as the number of slits goes to infinity. In the special case of n slits that connect n points of the boundary to one fixed point, one can take the limit of the Loewner equation that describes the growth of those slits in a simultaneous way. In this case, the limit is a deterministic Loewner equation whose vector field is determined by a complex Burgers equation.     

Matrix valued interpolation and truncated Hamburger moment problems

A solvability condition for matrix valued directional single-node interpolation problems of Loewner type is established, in terms of properties of Pick kernel. As a consequence, a solvability condition for matrix valued directional truncated Hamburger moment problems is obtained.     

Topological Loewner theory on Riemann surfaces

We prove that topological evolution families on a Riemann surface S are rather trivial unless S is conformally equivalent to the unit disc or the punctuated unit disc. We also prove that, except for the torus where there is no non-trivial continuous Loewner chain, there is a topological evolution family associated to any topological Loewner chain and, conversely, any topological evolution family comes from a topological Loewner chain on the same Riemann surface.     

Chordal Komatu–Loewner equation for a family of continuously growing hulls

In this paper, we discuss the chordal Komatu–Loewner equation on standard slit domains in a manner applicable not just to a simple curve but also a family of continuously growing hulls. Especially a conformally invariant characterization of the Komatu–Loewner evolution is obtained. As an application, we prove a sort of conformal invariance, or locality, of the stochastic Komatu–Loewner evolution ${\mathrm{SKLE}}_{\sqrt{6},?b_{\mathrm{BMD}}}$ in a fully general setting, which solves an open problem posed by Chen et al. (2017).     

On a numerical algorithm for the solution of the radial Loewner equation

The Loewner partial differential equation provides a one‐parametric family of conformal maps on the unit disk. The images describe a flow of an expanding simply‐connected domain, called the Loewner flow, on the complex plane. In this paper, we present a numerical algorithm for solving the radial Loewner partial differential equation. The algorithm is applied to visualization of Loewner flows with the performance and precision. From the theoretical point of view, our algorithm is based on a recursive formula for determining coefficients of polynomial approximations. We prove that each coefficient converges to true values with reasonable regularity.     

Loewner Transformations: Adjoint and Binary Darboux Connections

It is shown that a class of finite Bäcklund transformations introduced by Loewner in 1950 in a gasdynamics context may be represented as compound gauge and Darboux-type transformations. This result is used to construct iterated versions of the Loewner transformations based on established procedures in soliton theory.     

Chordal Loewner families and univalent Cauchy transforms

We study chordal Loewner families in the upper half-plane and show that they have a parametric representation. We show one, that to every chordal Loewner family there corresponds a unique measurable family of probability measures on the real line, and two, that to every measurable family of probability measures on the real line there corresponds a unique chordal Loewner family. In both cases the correspondence is being given by solving the chordal Loewner equation. We use this to show that any probability measure on the real line with finite variance and mean zero has univalent Cauchy transform if and only if it belongs to some chordal Loewner family. If the probability measure has compact support we give two further necessary and sufficient conditions for the univalence of the Cauchy transform, the first in terms of the transfinite diameter of the complement of the image domain of the reciprocal Cauchy transform, and the second in terms of moment inequalities corresponding to the Grunsky inequalities.     

The Power Matrix, coadjoint action and quadratic differentials

The coefficients of a quadratic differential which is changing under the Loewner flow satisfy a well-known differential system studied by Schiffer, Schaeffer and Spencer, and others. By work of Roth, this differential system can be interpreted as Hamilton's equations. We apply the power matrix to interpret this differential system in terms of the coadjoint action of the matrix group on the dual of its Lie algebra. As an application, we derive a set of integral invariants of Hamilton's equations which is in a certain sense complete. In function theoretic terms, these are expressions in the coefficients of the quadratic differential and Loewner map which are independent of the parameter in the Loewner flow.