Point selections from Jordan domains in Riemannian Surfaces

TL;DR

This paper develops a method for continuously selecting points within Jordan domains on Riemannian surfaces, ensuring equivariance under isometries and conformal maps, with applications to deformation retractions and canonical point selection.

Contribution

It introduces a new continuous, equivariant point selection process for Jordan domains on Riemannian surfaces, extending to conformal transformations and canonical planar procedures.

Findings

Existence of equivariant point selection functions

Deformation retraction of domain spaces onto round disks

Canonical point selection method for planar Jordan domains

Abstract

Using fiber bundle theory and conformal mappings, we continuously select a point from the interior of Jordan domains in Riemannian surfaces. This selection can be made equivariant under isometries, and take on prescribed values such as the center of mass when the domains are convex. Analogous results for conformal transformations are obtained as well. It follows that the space of Jordan domains in surfaces of constant curvature admits an isometrically equivariant strong deformation retraction onto the space of round disks. Finally we develop a canonical procedure for selecting points from planar Jordan domains.