Comments on the regularity of harmonic maps between singular spaces

Abstract

In this work we are going to establish Hölder continuity of harmonic maps from an open set $\hat{I}\text{c◯}$ in an $\mathrm{RCD}(K,N)$ space valued into a $\mathrm{CAT}(\hat{I}º)$ space, with the constraint that the image of $\hat{I}\text{c◯}$ 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.

Comments are welcome!