arXiv research

Study lifts plane mappings to Heisenberg group.

In this paper we study the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group. We prove that all of the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group are helices. Moreover, we obtain explicit parametric equations for non-geodesic non-null biharmon…

Study Gauss maps of surfaces in Heisenberg group using hyperbolic geometry.

Study counts and equidistributes rational points in quaternionic Heisenberg groups.

The affine-additive group is hyperbolic with a non-vanishing 4-capacity.

It is shown that parts of planes, helicoids and hyperbolic paraboloids are the only minimal surfaces ruled by geodesics in the three dimensional Riemannian Heisenberg group. It is also shown that they are the only surfaces in the three dimensional Heisenberg group whose mean curvature is zero with respect to both of th…

We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary num…

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient co…

Study minimizes CR surfaces in Heisenberg group with rotational symmetry.

We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometr…

We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively hyperbolic group structure and have one end. Applications include surface mapping clas…

Classifies Einstein metrics on R^4 with Heisenberg symmetry, finding incomplete Ricci-flat metrics and two complete negative-curvature examples.

In this note, we obtain the asymptotic estimate for the time derivative of the $Φ$-entropy in terms of the lower bound on the Bakry-Emery $Γ_2$ curvature. In the cases of Hyperbolic space and Heisenberg group, we show that the time derivative of the $Φ$-entropy is non-increasing, and we also get sharp asymptotic bound …

We study the Gauss map of minimal surfaces in the Heisenberg group $\mathrm{Nil}_3$ endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane $\mathbb{H}^2$. Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{…

The paper classifies metrics on Heisenberg group's cotangent bundle.

We use Korányi--Reimann complex cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of complex hyperbolic space and the first Heisenberg group.

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature if and only if the pc conformal curvature vanishes. I…

We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean spaces. In the first Heisenbeg group, endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection $π(E)$ (projection along the center of the Heisenberg group) almost surely equals $\min\{2,dim_H(E)\}$ and …

Sharp inequalities on Siegel domains and complex hyperbolic spaces established.

The paper classifies all left invariant metrics on complex hyperbolic space.

Researchers find limits on curvature of certain 3D solitons.

Proves quaternionic analog of Cartan's theorem and counts arithmetic chains.

New conformally Einstein metrics on Heisenberg group found.

Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.

Classifies geodetically convex sets and functions on Heisenberg group.

For the ``Hopf bundle'' $S^1\to S^{2n,1} \to {\mathbb H}^n$, horizontal lifts of simple closed curves are studied. Let $γ$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along $γ$ is given by $V(γ)=e^{λA(γ) i}$ where $A(γ)$ is …

Homogeneous magnetic paths found in Heisenberg space.

Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.

Find conditions for starshapedness of level sets in Heisenberg group.

Defines contact structures on Heisenberg groups for geometric interpretation.

Computes minimal polynomials for generalized Heisenberg groups.

The paper describes geodesics on a Kähler cone of the Heisenberg group.

Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.

In Heisenberg group, bisectors are spinal spheres with specific curvature.

Criterion for surfaces in Heisenberg group to be graphs using flat cones.

Translation surfaces in Heisenberg group classified by Gauss map determinant.

The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.

Study maps surface configurations to Heisenberg homologies for mapping class groups.

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 use some results on Levi-Civita connection and curvature tensor to present solutio…

Study shows only hyperplanes in Heisenberg groups have zero curvature.

Develops analysis of Hölder continuous mappings on Heisenberg groups.

Study on Killing magnetic curves in Heisenberg group geometry.

In this paper, we study the Ricci-Bourguignon flow on higher dimensional classical Heisenberg nilpotent Lie groups and construct a solution of this flow on Heisenberg and quaternion nilpotent Lie groups. In the end, we investigate the deformation of spectrum and length spectrum on compact nilmanifolds obtained of Heise…

Sharp stability of isometries on Heisenberg group proven.

Study local control in a 7D quaternionic Heisenberg group.

A translation surface in the Heisenberg group $\mathrm{Nil}_3$ is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group $\mathrm{Nil}_3$.

Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.