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 $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> in an $$\textrm{RCD}(K,N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>RCD</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> space valued into a $$\textrm{CAT}(\kappa )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>CAT</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>κ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> space, with the constraint that the image of $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> 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.