Measure contraction property on isometric leaves and monotone fibres
Abstract
For finite measures with positive densities on convex Euclidean supports, we prove that MCP(κ,N) passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.
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BibTeX
@misc{ciosmak2026measure,
title = {{Measure contraction property on isometric leaves and monotone fibres}},
author = {Krzysztof J. Ciosmak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21510},
eprint = {2609.21510},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21510},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Measure contraction property on isometric leaves and monotone fibres
AU - Krzysztof J. Ciosmak
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21510
UR - https://arxiv.org/abs/2609.21510
AB - For finite measures with positive densities on convex Euclidean supports, we prove that MCP(κ,N) passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.
ER -