Geometric obstructions to Lipschitz transport between weighted Hessian $\mathrm{CD}(\hat{I}º,\infty )$ manifolds

Abstract

We construct a weighted Riemannian manifold $({\mathbb{R}}^2,g,\hat{I}¼)$ satisfying $\mathrm{CD}(1/2,\infty )$, the curvature-dimension condition, with the following property: if ${\hat{I}}^3$ denotes a centered Gaussian measure on ${\mathbb{R}}^2$, then there is no Lipschitz map $T:({\mathbb{R}}^2,\Vert \cdot \Vert )\to ({\mathbb{R}}^2,g)$ satisfying $T_{\#}{\hat{I}}^3=\hat{I}¼$. Building on this, we prove a Weyl-type asymptotic law for the eigenvalues of the weighted Laplacian $-\hat{I}{”}_{g,\hat{I}¼}$ and show that they are asymptotically negligible when compared to the eigenvalues of $-\hat{I}{”}_{\hat{I}}^3$. These results give strong counterexamples to two questions of E. Milman and complement the recent counterexample of Aryan.

26 pages, 1 figure; new version: minor edits and improved exposition