53 problems

Let $\mathbb{G}$ be an $m$-dimensional Carnot group with horizontal distribution $\Delta=\operatorname{span}\{X_1,\ldots,X_k\}$, let $I=[0,1]$, and set $H=L^2(I,\mathbb{R}^k)$. For…

Dimension-free cut-locus structure conjecture. The cut locus has the form

Scaling conjecture. There is a map $\alpha:S^k\to G$ such that

A Carnot group is a graded nilpotent Lie group, and its subRiemannian geodesic flow is the geodesic flow associated with its subRiemannian structure. Complete integrability conject…

The study of Sobolev and Poincaré inequalities for differential forms in Carnot groups and, more generally, in sub-Riemannian settings remains open in full generality. Graded Lebes…

Direct-type criterion. The geodesic $\gamma(t)$ is of direct-type if and only if

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…

Metric-line classification conjecture. The metric lines in $\mathcal{J}^k(\mathbb{R},\mathbb{R}^n)$ are precisely the line, homoclinic, and direct-type geodesics.

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 $N_{\mathrm{GEO}}$…

For $0<\varphi_2<\varphi_1$, write $U_i=U(\varphi_i)$ and $Z_{ij}=Z(\varphi_i,\varphi_j)$ for the functions defined in the paper. Positivity conjecture. One has … for every…

Let $\mathbb F_4$ be the free step-2 rank-four Carnot group, and let $\mathcal H$ be the union of all strict Carnot subgroups. Equivalently, … Let…

Let $\mathbb F_n$ be the free step-2 rank-$n$ Carnot group, and let $\operatorname{Cut}(\mathbb F_n)$ denote its cut locus. Let $C_n\subset\mathbb F_n$ be the set defined in the so…

Let $\mathbb{G}$ be a Carnot group with horizontal distribution $\Delta$, equipped with a left-invariant norm, and let $H=L^2(I,\mathbb{R}^k)$ be the control space for horizontal p…

Let $\mathcal{L}$ be a nonelliptic sub-Laplacian on a Carnot group, and let $\varsigma_{\mathrm{wave}}(\mathcal{L})$ and $\varsigma_{\mathrm{mult}}(\mathcal{L})$ denote its wave-eq…

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…

Consider the geodesics of the Carnot group under study, parametrized by the re-scaled time $d70f$. The preceding analysis identifies $d70f=\pi$ as a Maxwell re-scaled time for geod…

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…

Let $\mathbb{G}$ be a step two Carnot group of rank $m$. The sharp H-type deviation bound conjecture. … The value $\delta(\mathbb{F}_{2,m})=\sqrt{(m-2)/m}$ for the free step two Ca…

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 $G$ be a stratified Banach-Lie group with finite-dimensional commutator subgroup. An Aronszajn null set is a set null on the relevant affine lines in the Aronszajn sense, and a…

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…