Spectral Obstructions to Contracting Transport Maps on Curved Spaces

Abstract

Caffarelli's contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is $1$-Lipschitz. Motivated by this theorem, Milman [Mil18] formulated several conjectures for the round sphere and for weighted manifolds satisfying the curvature-dimension condition $\operatorname{CD}(\rho,\infty)$. Recently, Beck and Jerison [BJ21] raised related questions on the round hemisphere. The existence of a contracting transport map implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi [CM98] for compact manifolds with Ricci curvature lower bounds. In this work, we construct counterexamples to the corresponding spectral comparisons on spheres and on weighted manifolds satisfying $\operatorname{CD}(1,\infty)$ in dimensions $d\geq4$, yielding obstructions to contracting transport maps. In dimensions $d\geq5$, the weighted counterexamples can be chosen to satisfy $\operatorname{Ric}_g\geq 0$ and $\nabla_g^2 V \geq g$ separately. In dimension two, we use inverse mean curvature flow to construct a contracting transport map from the suitably rescaled round sphere to every closed connected Riemannian surface satisfying the same positive Ricci curvature lower bound. This implies the spectral comparison in dimension two. Together with the recent counterexample in dimension three by Lin, Wang, and Xu [LWX26], this settles the spherical spectral comparison in every dimension $d\geq2$. Using the same method, we also construct a contracting transport from the uniform probability measure on a hemisphere onto the normalized uniform measure on any geodesically convex subset of positive volume, thereby answering affirmatively the remaining case of a conjecture by Beck and Jerison [BJ21] following the work of Fathi, Fradelizi, Gozlan, and Zugmeyer [FFGZ26].