Comments on the Regularity of Harmonic Maps Between Singular Spaces
Luca Gennaioli, Nicola Gigli, Hui-Chun Zhang, Xi-Ping Zhu · Journal of Geometric Analysis · 2026
Abstract In this work we are going to establish Hölder continuity of harmonic maps from an open set $$\Omega $$ Ω in an $$\textrm{RCD}(K,N)$$ RCD ( K , N ) space valued into a $$\textrm{CAT}(\kappa )$$ CAT ( κ ) space, with the constraint that the image of $$\Omega $$ Ω 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.