The measure contraction property on Grushin spaces

Abstract

We determine the sharp measure contraction exponents of two families of Grushin-type metric measure spaces. The radial Grushin space 𝔾^n+m is ℝⁿ×ℝ^m, equipped with Lebesgue measure and generated by X_i=∂_{x_i} and Y_j=|x|∂_{y_j}, for 1≤ i≤ n and 1≤ j≤ m. We prove that 𝔾^n+m satisfies MCP(K,N) if and only if N≥ n+4m and K≤ 0. We also show that, for α≥1, the α-Grushin plane generated by X=∂_x and Y_α=|x|^α∂_y satisfies MCP(K,N) if and only if K≤0 and N≥ N_α, where N_α:= 1+max_L>1 ((2α+1)L)/((L−1)^2α+1+1). This resolves the conjecture posed in arXiv:2010.16350 and, for integer α≥2, provides the first examples of real-analytic sub-Riemannian structures with noninteger curvature exponent. Both results recover the known curvature exponent $5$ of the classical Grushin plane, corresponding respectively to n=m=1 and α=1.

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BibTeX RIS

BibTeX

@misc{albert2026measure,
  title = {{The measure contraction property on Grushin spaces}},
  author = {Michael Albert and Samuël Borza},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10208},
  eprint = {2609.10208},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10208},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The measure contraction property on Grushin spaces
AU  - Michael Albert
AU  - Samuël Borza
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10208
UR  - https://arxiv.org/abs/2609.10208
AB  - We determine the sharp measure contraction exponents of two families of Grushin-type metric measure spaces. The radial Grushin space 𝔾^n+m is ℝ^n×ℝ^m, equipped with Lebesgue measure and generated by X_i=∂_x_i and Y_j=|x|∂_y_j, for 1≤ i≤ n and 1≤ j≤ m. We prove that 𝔾^n+m satisfies MCP(K,N) if and only if N≥ n+4m and K≤ 0. We also show that, for α≥1, the α-Grushin plane generated by X=∂_x and Y_α=|x|^α∂_y satisfies MCP(K,N) if and only if K≤0 and N≥ N_α, where N_α:= 1+max_L>1 ((2α+1)L)/((L−1)^2α+1+1). This resolves the conjecture posed in arXiv:2010.16350 and, for integer α≥2, provides the first examples of real-analytic sub-Riemannian structures with noninteger curvature exponent. Both results recover the known curvature exponent $5$ of the classical Grushin plane, corresponding respectively to n=m=1 and α=1.
ER  -

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