A Bakry-Émery Approach to Lipschitz Transportation on Manifolds

On weighted Riemannian manifolds we prove the existence of globally Lipschitz transport maps between the weight (probability) measure and log-Lipschitz perturbations of it, via Kim and Milman’s diffusion transport map, assuming that the curvature-dimension condition \(\varvec{\textrm{CD}(\rho _{1}, \infty )}\) $\text{CD}\mathbf{(}{\mathit{\rho }}_{\mathbf{1}}\mathbf{,}\mathit{\infty }\mathbf{)}$ holds, as well as a second order version of it, namely \(\varvec{\Gamma _{3} \ge \rho _{2} \Gamma _{2}}\) ${\mathbf{\Gamma }}_{\mathbf{3}}\mathbf{\ge }{\mathit{\rho }}_{\mathbf{2}}{\mathbf{\Gamma }}_{\mathbf{2}}$ . We get new results as corollaries to this result, as the preservation of Poincaré’s inequality for the exponential measure on \(\varvec{(0,+\infty )}\) $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{+}\mathit{\infty }\mathbf{)}$ when perturbed by a log-Lipschitz potential and a new growth estimate for the Monge map pushing forward the gamma distribution on \(\varvec{(0,+\infty )}\) $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{+}\mathit{\infty }\mathbf{)}$ (then getting as a particular case the exponential one), via Laguerre’s generator.