The curvature exponent–lower-bound conjecture for Carnot groups

Let $N_{\mathrm{CE}}$ be the curvature exponent of a Carnot group, namely the least $N$ for which the measure contraction property $\mathrm{MCP}(0,N)$ holds. Let $N_0$ be the lower bound for $N_{\mathrm{CE}}$ obtained from the small-time asymptotic behavior of the Jacobian of the sub-Riemannian exponential map. Curvature exponent–lower-bound conjecture. In the setting of Carnot groups with negligible cut loci, we have $N_{\mathrm{CE}} = N_0$. The conjecture was disproved by an ideal Carnot group for which $N_{\mathrm{CE}} > N_0$; consequently, $N_0$ is not lower semicontinuous.