Spectral Obstructions to Contracting Transport Maps on Curved Spaces

Abstract

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. Motivated by this theorem, Milman [Mil18] formulated several conjectures for the round sphere and for weighted manifolds satisfying the curvature-dimension condition CD(ρ,∞). A contracting transport map as in these conjectures implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi [CM98] for compact manifolds with Ricci curvature lower bounds. We construct counterexamples to these conjectured spectral comparisons on spheres in dimensions d≥4 and on smooth complete weighted manifolds diffeomorphic to ℝ^d satisfying the CD(1,∞) condition in dimensions d≥4, thereby obtaining obstructions to contracting transport maps. The spherical counterexamples can be chosen arbitrarily close to the unit round metric in C^∞, while retaining Ric_g≥(d−1)g. The weighted counterexamples can be chosen with non-negative sectional curvature Sec_g≥0 when d≥4. For d≥5, we also construct counterexamples satisfying Ric_g≥0 and ∇_g²V≥ g, where μ=Z⁻¹e^−Vdvol_g.

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BibTeX RIS

BibTeX

@misc{aryan2026spectral,
  title = {{Spectral Obstructions to Contracting Transport Maps on Curved Spaces}},
  author = {Shrey Aryan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.24705},
  eprint = {2605.24705},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.24705},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Spectral Obstructions to Contracting Transport Maps on Curved Spaces
AU  - Shrey Aryan
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2605.24705
UR  - https://arxiv.org/abs/2605.24705
AB  - 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. Motivated by this theorem, Milman [Mil18] formulated several conjectures for the round sphere and for weighted manifolds satisfying the curvature-dimension condition CD(ρ,∞). A contracting transport map as in these conjectures implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi [CM98] for compact manifolds with Ricci curvature lower bounds. We construct counterexamples to these conjectured spectral comparisons on spheres in dimensions d≥4 and on smooth complete weighted manifolds diffeomorphic to ℝ^d satisfying the CD(1,∞) condition in dimensions d≥4, thereby obtaining obstructions to contracting transport maps. The spherical counterexamples can be chosen arbitrarily close to the unit round metric in C^∞, while retaining Ric_g≥(d−1)g. The weighted counterexamples can be chosen with non-negative sectional curvature Sec_g≥0 when d≥4. For d≥5, we also construct counterexamples satisfying Ric_g≥0 and ∇_g^2V≥ g, where μ=Z^−1e^−Vdvol_g.
ER  -

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