Computes minimal polynomials for generalized Heisenberg groups.
problem None explicitly stated; focus on method.
method Computes minimal polynomials for generalized Heisenberg groups.
result Explicit minimal polynomials for generalized Heisenberg groups.
We classify connected Lie groups which are locally isomorphic to generalized Heisenberg groups. For a given generalized Heisenberg group N, there is a one-to-one correspondence between the set of isomorphism classes of connected Lie groups which are locally isomorphic to N and a union of certain quotients of noncom…
Develops analysis of Hölder continuous mappings on Heisenberg groups.
problem Analyzing Hölder continuous mappings on Heisenberg groups.
method Theory of distributional Jacobians and pullbacks of differential forms.
result Simple proof of a generalization of the Gromov non-embedding theorem and new results about Hölder homotopy groups.
Isomorphic algebra connects Toeplitz to Heisenberg group.
problem Connecting Toeplitz algebra to Heisenberg group.
method Isomorphism between Toeplitz algebra and Heisenberg group ideal.
result Found isomorphism between algebra and Heisenberg group ideal.
Minimal cylinders in Heisenberg group characterized using loop group method.
problem Characterizing minimal cylinders in the Heisenberg group.
method Generalized Weierstrass type representation and loop group method.
result Characterization of all non-vertical minimal cylinders in terms of pairs of closed plane curves.
New proof shows no Hölder embeddings into Heisenberg group.
problem Non-existence of Hölder embeddings into Heisenberg group.
method Developed a new elementary proof method.
result Generalization of Gromov's theorem proven.
Study timelike minimal surfaces in Heisenberg group using harmonic maps.
problem Characterize timelike minimal surfaces in the Heisenberg group.
method Characterization through non-conformal Lorentz harmonic maps and generalized Weierstrass type representation.
result Established generalized Weierstrass type representation for timelike minimal surfaces.
Unique minimal translation surfaces found in Heisenberg group.
problem Classifying minimal translation surfaces in Heisenberg group.
method Complete classification through generating curves analysis.
result Uniqueness of minimal translation surfaces up to isometries.
The paper classifies hypersurfaces in Heisenberg groups with rotational symmetry.
problem Classifying hypersurfaces in Heisenberg groups with rotational symmetry.
method Fundamental theorems and earlier results in [3] and [4] were used to classify umbilic hypersurfaces and generate curves for hypersurfaces with constant p-mean curvature. result Complete classification of umbilic hypersurfaces and generating curves in Heisenberg groups Hn. Maps in Carnot groups are equivalent to solutions of a PDE system.
problem Understanding maps in Carnot groups of step 2.
method Equivalence between intrinsic Lipschitz maps and solutions to a PDE system.
result Intrinsic Lipschitz maps are equivalent to weak solutions of a PDE system.
New conformally Einstein metrics on Heisenberg group found.
problem Finding new conformally Einstein metrics on specific Lie groups.
method Described Lorentzian semi-direct extensions of the Heisenberg group.
result Classified Bach-flat left-invariant Lorentzian metrics.
Study Heisenberg homology on surface configurations, revealing new representations of mapping class groups.
problem Homology of surface configurations with Heisenberg group representations.
method Analysis of unordered configurations in a surface, using Heisenberg group actions and representations.
result Obtained genuine and projective representations of mapping class groups from Heisenberg group actions.
Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.
problem Understanding submanifolds with boundary in sub-Riemannian Heisenberg groups.
method Introduced examples and proved Stokes' Theorem involving Rumin's differential forms.
result Stokes' Theorem for submanifolds with boundary in Heisenberg groups.
Classifies geodetically convex sets and functions on Heisenberg group.
problem Characterizing geodetically convex sets and functions in the Heisenberg group.
method Classification through mathematical analysis.
result Geodetically convex sets and functions defined on Heisenberg group Hn classified. Homogeneous magnetic paths found in Heisenberg space.
problem Understanding magnetic geodesics in the Heisenberg group.
method Proving homogeneity of magnetic geodesics derived from the canonical contact structure.
result Magnetic geodesics in the Heisenberg group are homogeneous.
The paper triangulates Heisenberg groups with horizontal and straight simplexes.
problem Triangulating Heisenberg groups with specific regularity properties.
method Constructing triangulations with horizontal and straight simplexes on a polyhedral structure and extending to the whole Heisenberg group.
result Explicit examples of grid and triangulations provided.
Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.
problem Understanding submanifolds with boundaries in Heisenberg groups.
method Introduce and study CH1-regular submanifolds with boundaries, prove Stokes' Theorem for them. result Prove a version of Stokes' Theorem for submanifolds with boundaries in Heisenberg groups.
Find conditions for starshapedness of level sets in Heisenberg group.
problem Ensure starshapedness of level sets of p-capacitary potentials. method Examine horizontally p-harmonic functions in the Heisenberg group. result Sharp conditions for strictly starshaped level sets.
We present a brief overview of the Korányi-Reimann theory of quasiconformal mappings on the Heisenberg group stressing on the analogies as well as on the differences between the Heisenberg group case and the classical two-dimensional case. We examine the extensions of the theory to more general spaces and we state some…
Defines contact structures on Heisenberg groups for geometric interpretation.
problem Finding a geometric interpretation for the Yamabe equation on Heisenberg groups.
method Defines contact structures of Heisenberg type, introduces a natural connection, and computes conformal scalar curvature.
result Establishes equivalence between contact Riemannian manifolds and contact structures of Heisenberg type.
Paper proves non-compact inaudibility of symmetry and commutativity.
problem Proving inaudibility of symmetry and commutativity in non-compact settings.
method Using isospectral pairs of generalized Heisenberg groups.
result Proved inaudibility of weak symmetry and commutativity.
The paper describes geodesics on a Kähler cone of the Heisenberg group.
problem Understanding geodesics on the Kähler cone of the Heisenberg group.
method Analyzing the Heisenberg group to describe geodesics.
result It is not a complete manifold.
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
In Heisenberg group, bisectors are spinal spheres with specific curvature.
problem Understanding bisectors in the Heisenberg group.
method Showed bisectors are spinal spheres and calculated their curvature.
result Metric bisectors in Heisenberg group are spinal spheres with specific curvature.
The paper studies metrics and geodesics on a quaternionic Heisenberg group.
problem Characterizing geodesics and distances on a quaternionic Heisenberg group.
method Defining and analyzing a sequence of Riemannian metrics, deriving formulas for mean curvature.
result Explicit description of Carnot-Carathéodory distance and spheres.
Criterion for surfaces in Heisenberg group to be graphs using flat cones.
problem Characterizing surfaces in Heisenberg group as graphs.
method Using planar cones to define intrinsic rectifiability.
result Criterion for topological surfaces to be intrinsic Lipschitz graphs.
Translation surfaces in Heisenberg group classified by Gauss map determinant.
problem Classifying translation surfaces with zero intrinsic curvature in Heisenberg group.
method Constructed surfaces as product of planar curves, classified by Gauss map determinant.
result Surfaces with vanishing intrinsic curvature identified.
We prove results toward classifying compact Lorentz manifolds on which Heisenberg groups act isometrically. We give a general construction, leading to a new example, of codimension-one actions--those for which the dimension of the Heisenberg group is one less than the dimension of the manifold. The main result is a cla…
Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
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 Heis3. 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.
problem Understanding Bernstein problem in higher dimensional Heisenberg groups.
method Sub-Riemannian characterization of ruling property and study of geodesics.
result Only hyperplanes have zero horizontal symmetric second fundamental form in Heisenberg groups.
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 on Killing magnetic curves in Heisenberg group geometry.
problem Understanding Killing magnetic curves in Heisenberg group.
method Presentation of Heisenberg group geometry and geodesics, study of Killing magnetic curves with explicit formulas.
result Explicit formulas for Killing magnetic curves in Heisenberg group.
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.
problem Quantitative stability of isometries on the Heisenberg group.
method Proving quasi-isometries close to isometries with specific closeness orders.
result Quasi-isometries of Heisenberg group are close to isometries with specific closeness orders.
Study local control in a 7D quaternionic Heisenberg group.
problem Optimizing geodesics in a 7D quaternionic Heisenberg group.
method Matrix representation and analysis of sub-Riemannian structure symmetries.
result Impact of symmetries on geodesic optimality.
A translation surface in the Heisenberg group Nil3 is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group Nil3.
The paper studies maps in the Heisenberg group and their images, called Rickman rugs.
problem Understanding maps and their images in the Heisenberg group.
method Analyzes maps f:WoH, where H is the first Heisenberg group and W is a vertical subgroup. result Rickman rugs in the Heisenberg group admit a corona decomposition by intrinsic bilipschitz graphs.
Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.
problem Determining the curvature exponent of sub-Finsler Heisenberg groups.
method Analyzing the measure contraction property and constructing sub-Finsler structures.
result Proved that curvature exponent N_min ≥ 5, with equality if sub-Riemannian.
Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.
problem Minkowski type problem in Heisenberg groups.
method Variational method.
result Positive answer to sub-Riemannian Minkowski type problem.
The paper studies sub-Riemannian geometry and proves a Weyl's invariance result for Heisenberg groups.
problem Optimal regularity and volume asymptotics of submanifolds in sub-Riemannian structures.
method Analyzes tubular neighborhoods and uses Weyl's invariance for Heisenberg groups.
result Volume of small tubes around curves in Heisenberg groups is invariant to embedding.
A controlled magnetic Hamiltonian (CMH) system is a regular controlled Hamiltonian (RCH) system with magnetic symplectic form, it is an important special case of RCH system. Note that there is a magnetic term on the cotangent bundle of the Heisenberg group, such that we can define a CMH system with symmetry of the Heis…
We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold M with a distribution H⊆TM analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this p…
The paper defines ASD connections and constructs families over a 5D Heisenberg group.
problem Defining and constructing ASD connections over a 5D Heisenberg group.
method Geometric approach using twistor spaces and Atiyah-Ward ansätz.
result Construction of families of ASD connections and their relation to vector bundles.
Study on Heisenberg group's Lorentzian problems using Pontryagin's principle.
problem Lorentzian problems on the Heisenberg group.
method Applied Pontryagin's maximum principle to obtain extremal trajectories.
result Parameterization of abnormal and normal extremal trajectories, investigation of reachability sets and existence of optimal trajectories.
Study eigenvalues and functions on specific Heisenberg manifolds.
problem Eigenvalues and eigenfunctions of a specific operator on Heisenberg Bieberbach manifolds.
method Analysis of eigenvalues and eigenfunctions of the Folland-Stein operator on Heisenberg Bieberbach manifolds.
result Characterized eigenvalues and eigenfunctions of the Folland-Stein operator on specific manifolds.
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in the Heisenberg group.
method Proves surfaces in a neighborhood of non-umbilic points are solutions to a sinh-Gordon equation with a differential constraint.
result Surfaces in Heisenberg group with constant mean curvature described by solutions to sinh-Gordon equation.
In Heisenberg groups, rectifiability is studied for subsets using C1,α-regular surfaces.
problem Understanding rectifiability of subsets in Heisenberg groups.
method Introducing a new notion of rectifiability and proving conditions for rectifiability using tangent paraboloids.
result A sufficient condition for C1,α-rectifiability of low-codimensional subsets in Heisenberg groups is the existence of suitable approximate tangent paraboloids.