Regularity of Mediatrices in Surfaces

TL;DR

This paper investigates the geometric regularity of mediatrices in two-dimensional Riemannian manifolds, revealing their radial linearizability and detailed structure on spheres, including singularities and angular deficiency.

Contribution

It establishes new regularity properties of mediatrices, such as radial linearizability, and characterizes their structure on spheres, including singularities and angular properties.

Findings

Mediatrices have the radial linearizability property.

On spheres, mediatrices are Lipschitz simple closed curves.

They have at most countably many singularities with finite total angular deficiency.

Abstract

For distinct points $p$ and $q$ in a two-dimensional Riemannian manifold, one defines their mediatrix $L_{pq}$ as the set of equidistant points to $p$ and $q$. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes additional geometric regularity properties of mediatrices. We show that mediatrices have the radial linearizability property, which implies that at each point they have a geometrically defined derivative in the branching directions. Also, we study the particular case of mediatrices on spheres, by showing that they are Lipschitz simple closed curves exhibiting at most countably many singularities, with finite total angular deficiency.