arXiv subjects
Publications and source records attributed to Oona Rainio.
Inclusion properties are studied for balls of the triangular ratio metric, the hyperbolic metric, the $j^*$-metric, and the distance ratio metric defined in the unit ball domain. Several sharp results are proven and a conjecture about the relation between triangular ratio metric balls and hyperbolic balls is given. An algorithm is also built for drawing triangular ratio circles or three-dimensional spheres.
Given a compact connected set $E$ in the unit disk $\mathbb{B}^{2}$, we give a new upper bound for the conformal capacity of the condenser $(\mathbb{B}^{2}, E)$ in terms of the hyperbolic diameter $t$ of $E$. Moreover, for $t>0$, we construct a set of hyperbolic diameter $t$ and apply novel numerical methods to show that it has larger capacity than a hyperbolic disk with the same diameter. The set we construct is called a Reuleaux triangle in hyperbolic geometry and it has constant hyperbolic width equal to $t$.
We study the conformal capacity by using novel computational algorithms based on implementations of the fast multipole method, and analytic techniques. Especially, we apply domain functionals to study the capacities of condensers $(G,E)$ where $G$ is a simply connected domain in the complex plane and $E$ is a compact subset of $G$. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of $E$. Our computational experiments demonstrate, for instance, sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.
We study numerical conformal mappings of planar Jordan domains with boundaries consisting of finitely many circular arcs and compute the moduli of quadrilaterals for these domains. Experimental error estimates are provided and, when possible, comparison to exact values or other methods are given. The main ingredients of the computation are boundary integral equations combined with the fast multipole method.
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.
For a non-empty compact set $E$ in a proper subdomain $Ω$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $Ω$ by $d(E)$ and $d(E,\partialΩ),$ respectively. The quantity $d(E)/d(E,\partialΩ)$ is invariant under similarities and plays an important role in Geometric Function Theory. In the present paper, when $Ω$ has the hyperbolic distance $h_Ω(z,w),$ we consider the infimum $κ(Ω)$ of the quantity $h_Ω(E)/\log(1+d(E)/d(E,\partialΩ))$ over compact subsets $E$ of $Ω$ with at least two points, where $h_Ω(E)$ stands for the hyperbolic diameter of the set $E.$ We denote the upper half-plane by $\mathbb{H}$. Our main results claim that $κ(Ω)$ is positive if and only if the boundary of $Ω$ is uniformly perfect and that the inequality $κ(Ω)\leqκ(\mathbb{H})$ holds for all $Ω,$ where equality holds precisely when $Ω$ is convex.
The distortion of six different intrinsic metrics and quasi-metrics under conformal and quasiregular mappings is studied in a few simple domains $G\subsetneq\mathbb{R}^n$. The already known inequalities between the hyperbolic metric and these intrinsic metrics for points $x,y$ in the unit ball $\mathbb{B}^n$ are improved by limiting the absolute values of the points $x,y$ and the new results are then used to study the conformal distortion of the intrinsic metrics. For the triangular ratio metric between two points $x,y\in\mathbb{B}^n$, the conformal distortion is bounded in terms of the hyperbolic midpoint and the hyperbolic distance of $x,y$. Furthermore, quasiregular and quasiconformal mappings are studied, and new sharp versions of the Schwarz lemma are introduced.
The triangular ratio metric is studied in a domain $G\subsetneq\mathbb{R}^n$, $n\geq2$. Several sharp bounds are proven for this metric, especially, in the case where the domain is the unit disk of the complex plane. The results are applied to study the Hölder continuity of quasiconformal mappings.
The point pair function $p_G$ defined in a domain $G\subsetneq\mathbb{R}^n$ is shown to be a quasi-metric and its other properties are studied. For a convex domain $G\subsetneq\mathbb{R}^n$, a new intrinsic quasi-metric called the function $w_G$ is introduced. Several sharp results are established for these two quasi-metrics, and their connection to the triangular ratio metric is studied.
A new intrinsic metric called $t$-metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains $G\subsetneq\mathbb{R}^n$. The behaviour of the new metric is also studied under a few examples of conformal and quasiconformal mappings, and the differences between the balls drawn with all the metrics considered are compared by both graphical and analytical means.
Hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under quasiconformal maps from one sector onto another one.