Measure contraction property conjecture for Grushin spaces

For each $\alpha\ge 1$ , let the $\alpha$ -Grushin plane be $\mathbb{R}^{2}=\mathbb{R}_{x}\times\mathbb{R}_{y}$ equipped with Lebesgue measure and the sub-Riemannian structure generated by $X=\partial_x$ and $Y_{\alpha}=|x|^{\alpha}\partial_y$ . Determine the sharp parameters $(K,N)$ for which it satisfies the measure contraction property $\operatorname{MCP}(K,N)$ . The conjectured classification is that $\operatorname{MCP}(K,N)$ holds if and only if $K\le 0$ and $N\ge N_{\alpha}$ , where $N_{\alpha}=1+\max_{L>1}\frac{(2\alpha+1)L}{(L-1)^{2\alpha+1}+1}$ .

References

Primary source

arXiv

Additional references

The measure contraction property on Grushin spaces — arXiv

metric-geometry sub-riemannian-geometry

Posted 4 weeks ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A September 2026 unrefereed preprint claims to settle the conjecture and extend the sharp threshold beyond integer cases, but the claim has not been independently verified.

The problem concerns the measure contraction property for Grushin spaces, including the broader $\alpha$ -Grushin family. The latest preprint claims a complete classification and new sharp examples.

Known results

The standard Grushin plane with Lebesgue measure satisfies $\mathrm{MCP}(K,N)$ if and only if $K\leq 0$ and $N\geq 5$ (2017).
Grushin half-planes satisfy $\mathrm{MCP}(K,N)$ if and only if $K\leq 0$ and $N\geq 4$ (2017).
A May 2025 preprint gave a computation-free proof of the standard Grushin-plane result via its quotient relation to the Heisenberg group.

September 2026 claimed resolution

A September 2026 arXiv preprint, The measure contraction property on Grushin spaces, claims necessary and sufficient $\mathrm{MCP}(K,N)$ conditions and an explicit sharp threshold for $\alpha$ -Grushin spaces, including noninteger curvature exponents. It therefore claims to resolve the conjecture, but remains unrefereed and unverified.

Current status (as of September 2026): The standard Grushin-plane and half-plane thresholds are established, while the broader $\alpha$ -Grushin classification is only claimed solved by an unrefereed preprint and remains unverified.

Sources

No solutions have been posted yet.