  We establish sufficient conditions for the existence of globally Lipschitz
transport maps between probability measures and their log-Lipschitz
perturbations, with dimension-free bounds. Our results include Gaussian
measures on Euclidean spaces and uniform measures on spheres as source
measures. More generally, we prove results for source measures on manifolds
satisfying strong curvature assumptions. These seem to be the first examples of
dimension-free Lipschitz transport maps in non-Euclidean settings, which are
moreover sharp on the sphere. We also present some applications to functional
inequalities, including a new dimension-free Gaussian isoperimetric inequality
for log-Lipschitz perturbations of the standard Gaussian measure. Our proofs
are based on the Langevin flow construction of transport maps of Kim and
Milman.
