Intrinsic metrics under conformal and quasiregular mappings

Intrinsic metrics in ring domains

Three hyperbolic type metrics including the triangular ratio metric, the $j^*$-metric and the Möbius metric are studied in an annular ring. The Euclidean midpoint rotation is introduced as a method to create upper and lower bounds for these metrics, and their sharp inequalities are found. A new Möbius-invariant lower bound is proved for the conformal capacity of a general ring domain by using a symmetric quantity defined with the Möbius metric.

Intrinsic metrics in polygonal domains

We study inequalities between the hyperbolic metric and intrinsic metrics in convex polygonal domains in the complex plane. Special attention is paid to the triangular ratio metric in rectangles. A local study leads to an investigation of the relationship between the conformal radius at an arbitrary point of a planar domain and the distance of the point to the boundary.

Conformal growth rates and spectral geometry on distributional limits of graphs

For a unimodular random graph $(G,ρ)$, we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of $(G,ρ)$, which is the best asymptotic degree of volume growth of balls that can be achieved by such a reweighting. Under moment conditions on the degree of the root, we show that the conformal growth exponent of a unimodular random graph bounds its almost sure spectral dimension. This has interesting consequences for many low-dimensional models. The consequences in dimension two are particularly strong…

内蕴Gromov双曲空间的一致化

Uniformization of intrinsic Gromov hyperbolic spaces

本文针对非测地、非紧合的Gromov双曲空间提出一致化方法，证明其共形形变为有界一致空间，且Gromov边界与形变空间度量边界存在自然拟等距，推广了已有经典结果。

无限正牛顿共形度量在角点附近的完备纤维

Completion Fibres of Infinite Positive Newton Conformal Metrics Near Corners

本文研究余维二角点附近正奇异共形度量的局部完备化，发现临界幂次时需拉普拉斯变换补充支撑几何，并给出纤维空或单点的判定及多种相关性质。

Relating Notions of Convergence in Geometric Analysis

We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal, $g_j=f_j^2g_0$, to a fixed manifold. We then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold $M$ are bounded, $(1-1/j)g_0 \le g_j \le K g_0$, and either the volumes converge, $\operatorname{Vol}_j(M)\right…

度量空间和洛伦兹预长度空间的共形变换

Conformal transformations of metric spaces and Lorentzian pre-length spaces

本文提出在度量空间和洛伦兹预长度空间中定义共形变换，并分析其性质，包括共形长度、角度不变性及因果结构的等价关系。

边界处的双曲畸变与共形性

Hyperbolic distortion and conformality at the boundary

本文用单位圆盘内蕴度量几何中的双曲畸变非切向渐近行为刻画全纯自映射在边界点的两类共形性，并给出算子论刻画及离散动力系统后向动力学中的应用。