43 problems

Let $H$ be the Kepler–Heisenberg Hamiltonian, and let $z(t)$ denote the vertical coordinate of a zero-energy orbit. An orbit avoids the $z$-axis if it never meets the set where the…

For $n\geq 2$, let $\triangle_1(\lambda)$ be the Laplacian on 1-forms in the representation with parameter $\lambda$, and let $U_{ij}$ be the transposition operators defined from t…

Let $(\mathbb H,d^{\mathbb H})$ be the Heisenberg group equipped with its Carnot–Carathéodory metric. Lee–Naor conjecture. The metric space $(\mathbb H,d^{\mathbb H})$ does not adm…

Spherical isoperimetric-surface conjecture. Isoperimetric surfaces in ${\mathrm{Nil}\,}$ are homeomorphic to spheres.

Isoperimetric-sphere conjecture. The constant-mean-curvature spheres of revolution in ${\mathrm{Nil}\,}$ are isoperimetric surfaces.

The Heisenberg Bernstein conjecture. Then $\mathcal S$ should be a vertical plane $\widetilde P_0$.

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

Let the bilinear tube estimate in Theorem $1$ be understood with its stated $L^p$ exponent, and let the sharp exponent mean the smallest exponent for which that estimate holds. Sha…

Work in the Heisenberg group $\mathbb{H}$, equipped with its standard metric and Hausdorff measure, and consider line segments lying entirely along the $x$- or $y$-axis. The dimens…

Let $\mathbb{H}$ be the first Heisenberg group, let $A\subseteq\mathbb{H}$ be a Borel set, and for $\theta\in[0,\pi)$ let $P_{\mathbb{V}_{\theta}^{\perp}}$ denote the vertical proj…

Let $S^{2n+1}$ be the CR sphere with standard contact form … The CR automorphisms of $S^{2n+1}$ are those induced by biholomorphisms of the unit ball in $\mathbb{C}^{n+1}$. Jerison…

Let $\mathbb{H}=\mathbb{C}\times\mathbb{R}$ be the first Heisenberg group, equipped with its Korányi metric, and let $\dim$ denote Hausdorff dimension with respect to this metric.…

Let $H$) be the Heisenberg group, and let $H\rtimes_{\psi}\mathbb{R}$ be its semidirect product with $\mathbb{R}$, where the action is given by … An embedding is $\varepsilon$-thic…

Let $u$ be the parameter defining the box $\operatorname{BP}_u$, let $W_u:=\operatorname{BP}_u\cap\\{(x,y,z):|x|\leqslant 1\\}$, and let $\partial W_u$ be its boundary. Let…

Let $u$ be the parameter defining the box $\operatorname{BP}_u$, let $W_u:=\operatorname{BP}_u\cap\\{(x,y,z):|x|\leqslant 1\\}$, and let $\partial W_u$ be its boundary. Let…

Positive lower-bound conjecture. There exists a positive lower bound for the operator inequality in (31); equivalently, there is a constant $A>0$ such that

Let $J$ denote the dilational momentum and consider a zero-energy orbit meeting the $z$-axis. Bifurcation conjecture. If … then the orbit has self-similarity like a generic $H=0$ o…

Let $H$ denote the Heisenberg group, and let contact harmonic graphs be the harmonic intrinsic graphs introduced in the paper. These graphs arise as limits of $H$-minimal surfaces…

Let $M$ be a sub-Riemannian manifold and let $\Sigma\subset M$ be an embedded hypersurface. Suppose that $\Sigma$ is at least $C^2$ and that its characteristic set is discrete. The…

Let $A$ be the averaging operator considered in Theorem 1.1, and let $I$ be the parameter interval. Assume that the condition denoted by (πLfold) holds for every $t\in I$, with no…

Vertical projection dimension conjecture. For almost every $\theta \in [0,\pi)$, if $0 \leq \dim A \leq 3$, then

Heisenberg-translate conjecture. For every nonzero $F\in L^2(\mathbb H)$, the collection

Carleson packing conjecture. Every intrinsic Lipschitz graph $\Gamma\subset\mathbb{H}$ satisfies this condition for $\eta\geq4$. The conjecture concerns whether vertical oscillatio…

Let $\Omega$ be an open subset of $\mathbb{R}^k$, let $n$ be a positive integer, and let $C^{\gamma}(\Omega,\mathbb{H}_n)$ denote the maps that are $\gamma$-Hölder with respect to…

Let $K$ be the best constant in the Heisenberg gradient estimates … for all $q\geqslant 1$ and $f\in C_{0}^{\infty}\left(\mathbb{G}\times\mathbb{R}^{3}\right)$, where $P_t$ is the…