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.