Related papers: Geodesics in the Heisenberg group

We prove quantitative stability of isometries on the first Heisenberg group with sub-Riemannian geometry: every $ (1+ \varepsilon)$-quasi-isometry of the John domain of the Heisenberg group $ \mathbb {H} $ is close to some isometry with…

We prove that a compact Sasakian manifolds whose first and second basic Chern classes vanish is locally isomorphic to the real Heisenberg group equipped with the standard left invariant Sasakian structure up to deformation associated to a…

Let G be a k-step Carnot group. We prove an isoperimetric-type inequality for compact C^2-smooth immersed hypersurfaces with boundary, involving the horizontal mean curvature of the hypersurface. This generalizes an inequality due to…

The non-trivial complete totally geodesic submanifolds of the complex hyperbolic plane $\mathbb H_{\mathbb C}^2$ are the complex geodesics and the real planes. We present two new proofs for this fact. One is a short proof based on an…

We establish unweighted Hardy-type inequalities on step-two Carnot groups with one-dimensional vertical layer, with explicit lower bounds for the optimal Hardy constant. The approach is based on a quantitative integration-by-parts mechanism…

We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $\sigma$-porous with respect to the Carnot-Carath\'eodory (CC) distance. In the first Heisenberg group we observe that there exist sets…

A categorification of the Heisenberg algebra is constructed in by Khovanov using graphical calculus, and left with a conjecture on the isomorphism between the Heisenberg algebra and Grothendieck ring of the constructed category. We give a…

The Riemann-Roch Theorem is one of the cornerstones of algebraic geometry, connecting algebraic data (sheaf cohomology) with geometric ones (intersection theory). This survey paper provides a self-contained introduction and a complete proof…

Collino proved that the fundamental group of a certain Zariski open set of the symmetric square of a hyperelliptic curve is isomorphic to the integral Heisenberg group. We compute the mixed Hodge structure on this fundamental group, and…

In this paper, we construct H\"older maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group $\mathbb{H}$. Pansu and Gromov observed that any surface embedded in $\mathbb{H}$ has Hausdorff dimension at…

A special type of coarea inequality is proved for compositions of intrinsically Lipschitz mappings of Carnot groups with projections along horizontal vector fields. It is proved that the equality is achieved for mappings with finite…

We compute the horizontal Hessian of the signed Carnot-Charatheodory distance from a surface S in the Heisenberg group H. The expression for the Hessian is in terms of the surface's intrinsic curvatures. As an application, we compute the…

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart…

We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space $H_{\mathbb R}^n$ endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the…

We prove the coarea formula for Lipschitz maps from the subriemannian $n$th Heisenberg group $\mathbb H_n$ to $\mathbb R^{2n}$. Our result is new even when $n=1$ and provides the simplest vector-valued instance of the coarea formula in…

Let $M$ be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order $2$. Generalizing results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques,…

Among eight possible geometric structures on three-dimensional manifolds less studied from the differential geometric point of view are those modelled on the Heisenberg group $Heis^3$. We consider the Heisenberg left-invariant metric and…

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\mathbb H^n$. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also…

We begin a systematic study of these spaces, initially following along the lines of Eberlein's comprehensive study of the Riemannian case. In particular, we integrate the geodesic equation, discuss the structure of the isometry group, and…

We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group…