Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps

TL;DR

This paper proves that convex functions on metric spaces, when composed with harmonic maps, produce subharmonic functions, extending known results to more general metric spaces and establishing maximum principles relevant to optimal transport.

Contribution

It generalizes the subharmonicity property of convex functions composed with harmonic maps to all metric spaces without curvature restrictions.

Findings

Convex functions composed with harmonic maps are subharmonic in general metric spaces.

Established maximum principles linking boundary and interior norms of composed functions.

Applied results to geodesically convex entropies in Wasserstein spaces, confirming conjectures.

Abstract

If $u : \Omega\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we prove that the pullback ${\sf E} \circ u : \Omega \to \mathbb{R}$ is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature restrictions on ${\rm X}$, while we prove it here in full generality. In addition, we establish generalized maximum principles, in the sense that the $L^q$ norm of ${\sf E} \circ u$ on $\partial \Omega$ controls the $L^p$ norm of ${\sf E} \circ u$ in $\Omega$ for some well-chosen exponents $p \geq q$, including the case $p=q=+\infty$. In particular, our results apply when ${\sf E}$ is a geodesically convex entropy over the Wasserstein space, and thus settle some conjectures of Y. Brenier, "Extended Monge-Kantorovich theory" in Optimal transportation and applications (Martina Franca, 2001), volume 1813 of Lecture Notes in Math., pages 91-121. Springer, Berlin, 2003.