28 problems

In the Martinet case, the cut-locus is the set of endpoints where minimizing geodesics cease to be minimizing; in the generic case, the northern hemisphere and the end-point of the…

Let an SR sphere mean a sphere for a sub-Riemannian structure, and call the case non-integrable when the associated geodesic dynamics is not integrable. Il'Yashenko's category is t…

Let $\beta$ be the parameter appearing in the non-integrable SR Martinet problem, and let an SR Martinet sphere denote the corresponding sub-Riemannian sphere. A function is pfaffi…

Distance-realization property. Then the distance $\operatorname{dist}(P,S_R)$ is realized by the point

Asymptotic-velocity conjecture. The metric lines in $\mathbb{G}$ are precisely the sub-Riemannian geodesics $\gamma(t)$ parameterized by arc length for which there exists a unit…

Cut-time conjecture. The cut time $t_{\operatorname{cut}}$ is identically

Failure of the measure contraction property with strong Goh–Legendre geodesics. If there is a strong-Goh–Legendre geodesic $\gamma$, then $\mathsf{d}^2:M\times M\to\mathbb{R}_+$ fa…

A sub-Finsler Carnot group is a stratified, nilpotent Lie group equipped with a left-invariant distance. Its homogeneous dimension is denoted by $\mathcal{Q}$, and…

Let $N_{\mathrm{CE}}$ denote the curvature exponent of a step-two Carnot group, namely the least $N$ for which the measure contraction property $\mathrm{MCP}(0,N)$ holds. Nonintegr…

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…

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 $N_{\mathrm{GEO}}$…

Let $(M,D,g)$ be a three-dimensional contact manifold with sub-Riemannian Laplacian $\Delta_{sR}$ and Reeb flow associated with the canonical contact form determined by $(D,g)$. A…

A sub-Riemannian diffusion killed when it exits a sufficiently small metric ball is expected to exhibit an analogous small-ball spectral convergence to the corresponding nilpotent…

Let $\mathbb{G}$ be a Carnot group with left-invariant sub-Riemannian structure. A curve $\gamma(t):\mathbb{R}\to\mathbb{G}$ is a metric line if it is a globally minimizing sub-Rie…

Eisenhart-type conjecture. If a sub-Riemannian metric $g$ is not affinely rigid near a point $q_0$, meaning that it admits a locally affinely equivalent non-constantly proportional…

Let $G$ be a metabelian Carnot group, let $A\triangleleft G$ be an abelian subgroup with $[G,G]\subseteq A$, and let $V_1$ denote the first layer of $\operatorname{Lie}(G)$. Assume…

Let $J^k(\mathbb{R},\mathbb{R})$ be the space of $k$-jets of real functions of one real variable, equipped with its sub-Riemannian distance. A geodesic…

Le Donne's conjecture. Carnot groups do not have periodic geodesics.

Curvature-dimension conjecture. For $b1 geqslant 1$, the $b1$-Grushin plane satisfies $\mathrm{MCP}(K,N)$ if and only if $K\leqslant 0$ and

Let $\phi$ be a norm of class $\mathrm{C}^\infty_+$, and let $\phi$-isoperimetric sets be isoperimetric sets for the norm $\phi$ in $\mathbb{H}^1$. Smooth regularity conjecture. Ev…

Cut-time equality conjecture. For any $\lambda \in C$

Horizontal diameter rigidity conjecture. For any singular Riemannian foliation on a unit sphere $\mathbb{S}^{n}$, we have

Let $G$ be a Carnot group with first layer $V_1$, let $\pi:G\to G/[G,G]$ be the quotient map, and let $d$ denote the relevant distance. Suppose that $\gamma:\mathbb{R}\to G$ is a g…

Let $H(\mathbb R)$ be the Heisenberg group with a Carnot–Carathéodory metric arising from a norm on the horizontal plane. In the sub-Riemannian case, Pansu's bubble set is the uniq…

Let $H(\mathbb R)$ be the Heisenberg group with its sub-Riemannian Carnot–Carathéodory metric. Let $S$ be a surface enclosing a region $E$, let $\lambda$ denote Haar measure, and l…