On nonhomeomorphic mappings between Riemannian manifolds

TL;DR

This paper investigates the properties of certain mappings between Riemannian manifolds, focusing on distortion estimates, extension to boundary points, and generalizations to metric spaces including Loewner spaces.

Contribution

It introduces new logarithmic estimates for distance distortion and explores conditions for continuous extension of mappings to boundary points, including in metric space settings.

Findings

Established logarithmic distance distortion estimates.

Proved conditions for continuous extension to boundary points.

Analyzed mappings in Loewner spaces and their equicontinuity.

Abstract

We consider mappings of domains of Riemannian manifolds that admit branch points and satisfy a certain condition regarding the distortion of the modulus of families of paths. We have established logarithmic estimates of distance distortion under such mappings. A separate study relates to the situation when the mappings are defined in metric spaces, and one of them is the Loewner space. We also studied the question of equicontinuity of the families of the indicated mappings in the closure of the domain. In addition, we have established the possibility of continuous extension of these mappings to an isolated point of the boundary.