9 problems

Curvature-exponent conjecture. Among all sub-Finsler Heisenberg groups, only the sub-Riemannian one has

Let $\mathbb G$ be a sub-Finsler Carnot group, and let its metric spheres be the level sets of the Carnot–Carathéodory distance associated with a sub-Finsler structure. Euclidean r…

Let $\mathbb G$ be a sub-Finsler Carnot group of step $2$, with Carnot–Carathéodory distance $d_{\mathrm{cc}}$, abnormal set $\mathrm{Abn}$, and a fixed Riemannian metric…

Sub-Finsler Carnot group curvature-dimension conjecture. The metric measure space $(G,\mathsf d_{SF},\mathfrak m)$ does not satisfy the $\mathsf{CD}(K,N)$ condition for any…

Curvature exponent five conjecture. The metric measure space $(\mathbb{H},\mathsf d,\mathscr{L}^3)$ satisfies $\mathsf{MCP}(0,5)$ if and only if the reference norm is the $\ell^2$-…

Classification conjecture. The functions $K_{\varepsilon,\sigma_0}$ are the only nontrivial positive solutions of this equation, and the best constant in the corresponding Sobolev…

Failure of the curvature-dimension condition conjecture. The metric measure space $(G,\mathsf d_{SF},\mathfrak m)$ does not satisfy the $\mathsf{CD}(K,N)$ condition for any…

Let $\mathbb{H}^n$ be a Heisenberg group equipped with a homogeneous distance $d$, and let $N$ be the underlying Euclidean norm associated to $d$. The space $(\mathbb{H}^n,d)$ has…

Let $(H(\mathbb R),d_Q,\lambda)$ be the Heisenberg group equipped with Haar measure $\lambda$ and an arbitrary Carnot–Carathéodory metric $d_Q$ arising from a norm on the horizonta…