Spectral Obstructions to Contracting Transport Maps on Curved Spaces
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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. Owing to its many applications in analysis, probability, and geometry, the problem of extending this theorem to curved spaces has appeared repeatedly in the literature, going back to Villani [V09]. More recently, Milman [Mil18] formulated precise conjectures in this direction. In this work, we construct counterexamples to these conjectures.
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Geometric obstructions to Lipschitz transport between weighted Hessian $\mathrm{CD}(\kappa,\infty)$ manifolds
Constructs CD(1/2,∞) manifold on R² without Lipschitz transport from centered Gaussian and proves its weighted Laplacian eigenvalues are asymptotically negligible compared to the Gaussian case.
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