# Korevaar-Schoen's directional energy and Ambrosio's regular Lagrangian flows

## TL;DR

This paper extends Korevaar-Schoen's directional energy theory to ${
m RCD}$ spaces, utilizing Ambrosio's Regular Lagrangian Flows, and offers new insights into differentials and identities in metric-valued Sobolev maps.

## Contribution

It develops a new framework for directional energies on ${
m RCD}$ spaces, integrating Ambrosio's flows and revisiting foundational concepts with novel stability results.

## Key findings

- Extended Korevaar-Schoen's theory to ${
m RCD}$ spaces
- Provided new insights into differentials along vector fields
- Established a new stability result for Regular Lagrangian Flows

## Abstract

We develop Korevaar-Schoen's theory of directional energies for metric-valued Sobolev maps in the case of ${\sf RCD}$ source spaces; to do so we crucially rely on Ambrosio's concept of Regular Lagrangian Flow. Our review of Korevaar-Schoen's spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of `differential of a map along a vector field' and about the parallelogram identity for ${\sf CAT}(0)$ targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows.