# Uniformization of metric surfaces

Main funder

Funds granted by main funder (€)

- 25 000,00

Project timetable

Project start date: 01/12/2021

Project end date: 30/11/2022

Summary

We are interested in when a metric surface is quasiconformally equivalent to a Riemannian surface. A metric

surface is a metric space with locally finite Hausdorff 2-measure homeomorphic to a connected 2-manifold, and

quasiconformal maps are homeomorphisms that distort areas and lengths of paths in a comparable manner.

For example, if the Hausdorff 2-measure of a ball of radius r is comparable to r^2, such a surface admits a

quasiconformal parametrization. Indeed, in this setting, it is possible to construct a metric analog of the Riemann

map taking a given quadrilateral to the Euclidean unit disk (Rajala, 2017). Such a surface then admits a natural

smooth Riemannian structure quasiconformally equivalent to the original metric surface (Ikonen, 2019).

Rajala's construction of the Riemann map starts by first constructing a harmonic function u obtaining the boundary

values zero and one, respectively, on two disjoint boundary arcs of the quadrilateral. Then Rajala constructs a

"harmonic conjugate" v such that f = (u,v) is a quasiconformal embedding.

We wish to investigate the existence of a quasiconformal embedding f = (u,v) in a related setting. For example, if the

measure of balls is not controlled by the square of the radius, but X supports a Poincaré inequality, is it possible to

construct the quasiconformal embedding f = (u,v) possibly by modifying the definitions of u and v? This is related to

the existence of a "Hodge star operator" on such a metric space. We are interested in this case, since many tools

from classical potential analysis are available.

surface is a metric space with locally finite Hausdorff 2-measure homeomorphic to a connected 2-manifold, and

quasiconformal maps are homeomorphisms that distort areas and lengths of paths in a comparable manner.

For example, if the Hausdorff 2-measure of a ball of radius r is comparable to r^2, such a surface admits a

quasiconformal parametrization. Indeed, in this setting, it is possible to construct a metric analog of the Riemann

map taking a given quadrilateral to the Euclidean unit disk (Rajala, 2017). Such a surface then admits a natural

smooth Riemannian structure quasiconformally equivalent to the original metric surface (Ikonen, 2019).

Rajala's construction of the Riemann map starts by first constructing a harmonic function u obtaining the boundary

values zero and one, respectively, on two disjoint boundary arcs of the quadrilateral. Then Rajala constructs a

"harmonic conjugate" v such that f = (u,v) is a quasiconformal embedding.

We wish to investigate the existence of a quasiconformal embedding f = (u,v) in a related setting. For example, if the

measure of balls is not controlled by the square of the radius, but X supports a Poincaré inequality, is it possible to

construct the quasiconformal embedding f = (u,v) possibly by modifying the definitions of u and v? This is related to

the existence of a "Hodge star operator" on such a metric space. We are interested in this case, since many tools

from classical potential analysis are available.