Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from RCD( K,N ) spaces to CAT(0) spaces

abstract: We establish Lipschitz regularity of harmonic maps from $\operatorname{RCD}(K,N)$ metric measure spaces with lower Ricci curvature bounds and dimension upper bounds in synthetic sense with values into $\mathsf{CAT}(0)$ metric spaces with non-positive sectional curvature. Under the same assumptions, we obtain a Bochner-Eells-Sampson inequality with a Hessian type-term. This gives a fairly complete generalization of the classical theory for smooth source and target spaces to their natural synthetic counterparts and an affirmative answer to a question raised several times in the recent literature.