Geometric obstructions to Lipschitz transport between weighted Hessian $\mathrm{C}\mathrm{D}(\kappa ,\mathrm{\infty })$ manifolds

We construct a weighted Riemannian manifold $({\mathbb{R}}^2,g,\mu )$ satisfying $\mathrm{C}\mathrm{D}(1\mathrm{/}2,\mathrm{\infty })$, the curvature-dimension condition, with the following property: if $\gamma$ denotes a centered Gaussian measure on ${\mathbb{R}}^2$, then every map $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ satisfying $T_{\mathrm{\#}}\gamma =\mu$ fails to be Lipschitz as a map from $({\mathbb{R}}^2,\mathrm{\parallel }\cdot \mathrm{\parallel })$ to $({\mathbb{R}}^2,g)$. Building on this, we prove a Weyl asymptotic law for the eigenvalues of the weighted Laplacian $-{\mathrm{\Delta }}_{g,\mu }$ and show that they are asymptotically negligible when compared to the eigenvalues of $-{\mathrm{\Delta }}_{g,\gamma }$. These results give strong counterexamples to two questions of E. Milman and complement the recent counterexample of Aryan.