# Hyperbolic type metrics in geometric function theory

##### Hariri, Parisa (2018-03-17)

Hariri, Parisa

Turun yliopisto

17.03.2018

**Julkaisun pysyvä osoite on:**

https://urn.fi/URN:ISBN:978-951-29-7159-6

##### Tiivistelmä

The research area of this thesis is Geometric Function Theory, which is a subfield of mathematical analysis. The thesis consists of four published papers. Prepublication versions of these papers are available on the www-pages of the arXiv.org preprint server. The objects of this research are subdomains of the Euclidean n-dimensional space, their geometries, and the function classes de_ned on these subdomains. Some examples of these function classes are conformal maps, analytic functions, and Möbius transformations in the plane case (n = 2), and in the higherdimensional case (n ≥ 2) bilipschitz, quasiconformal and quasiregular maps. The main results concern the moduli of continuity of the aforementioned classes of functions with respect to so called hyperbolic type geometries. As a model or ideal we have the classical hyperbolic or non-Euclidean geometry of the unit disk, which was discovered two centuries ago and is invariant under conformal mappings.

During the past 30 years, the research on these questions has been, and continues to be, very active and it has turned out that in dimensions n ≥ 3 one must give up the full invariance property and replace it with a weaker, quasi-invariance property. The special role of the boundary of the domain is a key feature of hyperbolic type geometries. Each hyperbolic type geometry is based on a specific notion of distance between two points, so called metric. A hyperbolic type metric between two points takes into account, in addition to the position of the points with respect to each other, also the position of the points with respect to the boundary of the domain.

In the first paper, we study the triangular ratio metric and compare it with some other hyperbolic type metrics. Moreover, we prove that quasiregular mappings are Hölder continuous with respect to the triangular ratio metric.

In the second paper, we have a similar aim but this time for the visual angle metric. We show that this metric is comparable to the triangular ratio metric in convex domains and prove that quasiconformal maps are uniformly continuous with respect to the visual angle metric.

In the third paper, we find sufficient conditions on the domains for which two above mentioned metrics are comparable. We also show that bilipschitz maps with respect to the triangular ratio metric are quasiconformal.

In the fourth paper we introduce a new hyperbolic type metric. We show that this metric is comparable to other metrics in so called uniform domains.

The results of this thesis have already found applications also in the works of other researchers. On the basis of these facts, one can say that the results shed new light on Geometric Function Theory. In addition to this, several open problems are formulated in this thesis.

During the past 30 years, the research on these questions has been, and continues to be, very active and it has turned out that in dimensions n ≥ 3 one must give up the full invariance property and replace it with a weaker, quasi-invariance property. The special role of the boundary of the domain is a key feature of hyperbolic type geometries. Each hyperbolic type geometry is based on a specific notion of distance between two points, so called metric. A hyperbolic type metric between two points takes into account, in addition to the position of the points with respect to each other, also the position of the points with respect to the boundary of the domain.

In the first paper, we study the triangular ratio metric and compare it with some other hyperbolic type metrics. Moreover, we prove that quasiregular mappings are Hölder continuous with respect to the triangular ratio metric.

In the second paper, we have a similar aim but this time for the visual angle metric. We show that this metric is comparable to the triangular ratio metric in convex domains and prove that quasiconformal maps are uniformly continuous with respect to the visual angle metric.

In the third paper, we find sufficient conditions on the domains for which two above mentioned metrics are comparable. We also show that bilipschitz maps with respect to the triangular ratio metric are quasiconformal.

In the fourth paper we introduce a new hyperbolic type metric. We show that this metric is comparable to other metrics in so called uniform domains.

The results of this thesis have already found applications also in the works of other researchers. On the basis of these facts, one can say that the results shed new light on Geometric Function Theory. In addition to this, several open problems are formulated in this thesis.

##### Kokoelmat

- Väitöskirjat [2295]