Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group

We prove that no curvature-dimension bound CD(K,N) holds in any Heisenberg group $\mathbb H_n$. On the contrary, the measure contraction property MCP(0, 2n + 3) holds and is optimal for the dimension 2n + 3. For the nonexistence of a curvature-dimension bound, we prove that the generalized “geodesic” Brunn–Minkowski inequality is false in $\mathbb H_n$. We also show in a new and direct way (and for all $n\in\mathbb N\backslash\{0\}$), that the general “multiplicative” Brunn–Minkowski inequality with dimension N > 2n + 1 is false.