Published Paper

Inserted: 4 feb 2022
Last Updated: 7 apr 2026

Journal: Amer. J. Math.
Volume: 148
Number: 2
Pages: 505--568
Year: 2022
Doi: https://doi.org/10.1353/ajm.2026.a986599

Abstract:

We establish Lipschitz regularity of harmonic maps from $\mathrm{RCD}(K,N)$ metric measure spaces with lower Ricci curvature bounds and dimension upper bounds in synthetic sense with values into $\mathrm{CAT}(0)$ 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.\\ The proofs build on a new interpretation of the interplay between Optimal Transport and the Heat Flow on the source space and on an original perturbation argument in the spirit of the viscosity theory of PDEs.

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