# Absolutely continuous curves in Finsler-like spaces

## TL;DR

This paper investigates absolutely continuous curves in Finsler-like spaces, establishing their properties, relations to gradient flows, and characterizations in Wasserstein spaces, including nonsmooth Finsler-like spaces, highlighting differences from symmetric cases.

## Contribution

It provides new results on the coincidence of different types of absolutely continuous curves in Finsler spaces and characterizes these curves in Wasserstein spaces, extending to nonsmooth Finsler-like spaces.

## Key findings

- Coincidence of three types of absolutely continuous curves in Finsler manifolds
- Existence and regularity of gradient flows in Finsler spaces
- Characterization of absolutely continuous curves in Wasserstein spaces

## Abstract

The present paper is devoted to the investigation of absolutely continuous curves in asymmetric metric spaces induced by Finsler structures. Firstly, for asymmetric spaces induced by Finsler manifolds, we show that three different kinds of absolutely continuous curves coincide when their domains are bounded closed intervals. As an application, a universal existence and regularity theorem for gradient flow is obtained in the Finsler setting. Secondly, we study absolutely continuous curves in Wasserstein spaces over Finsler manifolds and establish the Lisini structure theorem in this setting, which characterize the nature of absolutely continuous curves in Wasserstein spaces in terms of dynamical transference plans concentrated on absolutely continuous curves in base Finsler manifolds. Besides, a close relation between continuity equations and absolutely continuous curves in Wasserstein spaces is founded. Last but not least, we also consider nonsmooth ``Finlser-like" spaces, in which case most of the aforementioned results remain valid. Various model examples are constructed in this paper, which point out genuine differences between the asymmetric and symmetric settings.