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

TL;DR

This paper proves Lipschitz regularity and a Bochner-Eells-Sampson inequality for harmonic maps from synthetic Ricci curvature bounded spaces to non-positively curved metric spaces, extending classical smooth theory.

Contribution

It introduces a synthetic framework for harmonic maps from RCD spaces to CAT(0) spaces, generalizing classical results to non-smooth settings.

Findings

Harmonic maps are Lipschitz continuous under the given conditions.

Established a Bochner-Eells-Sampson inequality with a Hessian-type term.

Provided a positive answer to a longstanding open question in the field.

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.