# Polyhedral approximation and uniformization of metric surfaces

# Polyhedral approximation and uniformization of metric surfaces

The classical uniformization theorem states that every simply connected Riemannian 2-manifold is conformally equivalent to the disk, the plane or the 2-sphere. The uniformization theorem has since been extended to various metric space settings. In this talk, we give a solution to the uniformization problem for metric surfaces assuming only locally finite (Hausdorff) 2-measure. This generalizes previous results on the topic due to Bonk–Kleiner, Lytchak–Wenger and Rajala. Our proof is based on a new polyhedral approximation scheme for metric surfaces with locally finite 2-measure: any such surface is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry in a precise way. This result, in turn, is based on a general triangulation theorem for metric surfaces of independent interest.

This talk covers joint work with Paul Creutz and Dimitrios Ntalampekos.