Lipschitz continuity of harmonic maps between $\mathrm{RCD}(K,N)$ spaces and $\mathrm{CAT}(\kappa )$ spaces

Abstract

We are going to prove that energy minimizing harmonic maps from a domain $\Omega$ inside an $\mathrm{RCD}(K,N)$ whose image lies in a small ball inside a $\mathrm{CAT}(\kappa )$ space are locally Lipschitz continuous. This completes the picture concerning regularity of harmonic maps between singular spaces and justifies a variant of the Bochner-Eells-Sampson inequality between singular spaces.

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