Uniformization of metric surfaces


Main funder


Funds granted by main funder (€)

  • 25 000,00


Project timetable

Project start date01/12/2021

Project end date30/11/2022


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.


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Last updated on 2024-14-10 at 14:14