Comments on the regularity of harmonic maps between singular spaces

TL;DR

This paper proves H"older continuity and boundary regularity for harmonic maps between non-smooth metric spaces, extending classical results to ${ m RCD}(K,N)$ and ${ m CAT}( appa)$ spaces with new inequalities.

Contribution

It establishes regularity results and a weak Bochner-Eells-Sampson inequality for harmonic maps in non-smooth metric space settings.

Findings

H"older continuity of harmonic maps under certain conditions

Weak Bochner-Eells-Sampson inequality in non-smooth spaces

Boundary regularity results for harmonic maps

Abstract

In this work we are going to establish H\"older continuity of harmonic maps from an open set $\Omega$ in an ${\rm RCD}(K,N)$ space valued into a ${\rm CAT}(\kappa)$ space, with the constraint that the image of $\Omega$ via the map is contained in a sufficiently small ball in the target. Building on top of this regularity and assuming a local Lipschitz regularity of the map, we establish a weak version of the Bochner-Eells-Sampson inequality in such a non-smooth setting. Finally we study the boundary regularity of such maps.