34 problems

Carnot tangent characterization. The following are equivalent:

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

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…

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

Let $\mathbb{G}$ 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…

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…

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…

Let $\mathbb{G}$ be a Carnot group of homogeneous dimension $Q\geq 3$, and let $u$ be Folland's fundamental solution for the sub-Laplacian. The group is polarizable if…

Let $\mathbb{G}$ be a step two Carnot group with horizontal metric $g_h$. Denote by $\Sigma$ the horizontal layer in exponential coordinates, … Let $u$ be the fundamental solution…

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.

Let $G$ be a Carnot group other than $\mathbb{R}^n$ or the Heisenberg group $\mathbb{H}_n$ for any $n$, and let $U,U'$ be open subsets of $G$. Let $f:U\to U'$ be a quasiconformal h…

Let $G$supset U$and$U' ormal?" Wait. Cannot introduce invalid notation. Let $G$ be a rigid Carnot group, and let $U,U' e othing$ be open subsets of $G$. A quasiconformal homeomor…

Let $G$ be a Carnot group, and let $U,U'\subset G$ be open sets. Xie's conjecture. If $G$ is neither $\mathbb{R}^n$ nor a Heisenberg group $\mathbb{H}_n$, then every quasiconformal…

Let $G$ be a Carnot group. A group is rigid if the space of smooth contact embeddings $G\supset U\to G$ is finite-dimensional for every connected open subset $U\subset G$. Regulari…

Let ${\mathbb G}$ be a Carnot group and let $W$ be a horizontal half-space. Denote by $S_W$ the semigroup generated by $W$, by $\bar S_W$ its closure, by $\mathfrak w_{\bar S_W}$ t…

Markov convexity threshold conjecture. $G$ is not Markov $p$-convex for every

First-layer generation conjecture. The first layer $V_1(G)$ generates $G$.