The paper defines H-orientability for surfaces in the Heisenberg group and finds non-H-orientable surfaces.
problem Defining orientability in the Heisenberg group for surfaces.
method Defined H-orientability for H-regular 1-codimensional surfaces in Hn. result Existence of non-H-orientable H-regular surfaces in H1. Study on Rumin cohomology and Heisenberg orientability in Heisenberg group.
problem Analyzing Rumin cohomology and Heisenberg orientability in Heisenberg group.
method Careful description of Rumin cohomology, commutation of differential operators, pushforward and pullback definitions, and definition of Heisenberg orientability.
result Existence of Heisenberg regular non-Heisenberg orientable surfaces.
This note describes a canonical way to orient the Heisenberg 3-manifold.
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.
The study proves surfaces in a specific Heisenberg group must be simple planes.
problem Characterizing surfaces in a sub-Finsler Heisenberg group.
method Analyzes (X,Y)-Lipschitz surfaces in H1 with a sub-Finsler structure. result Complete, oriented, stable (X,Y)-Lipschitz surfaces are vertical planes. We prove that any C2 complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group H1 is either a Euclidean plane or congruent to the hyperbolic paraboloid t=xy.
The paper classifies surfaces in the Heisenberg space invariant under specific isometries.
problem Classifying surfaces in the Heisenberg space with specific geometric properties.
method Analyzing surfaces with mean curvature H=⟨N,∂zangle+λ under left-translations, rotations, and helicoidal motions. result Classification of λ-translators invariant under specific isometries. 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.
We identify the space of left-invariant oriented complex structures on the complex Heisenberg group, and prove that it has the homotopy type of the disjoint union of a point and a 2-sphere.
We show that if the lower central series of the fundamental group of a closed oriented 3-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion 2-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental group…
We construct compact arbitrary Euler characteristic orientable and non-orientable minimal surfaces in the Berger spheres. Besides we show an interesting family of surfaces that are minimal in every Berger sphere, characterizing them by this property. Finally we construct, via the Daniel correspondence, new examples of …
Invariant of 3-manifolds using Hopf algebra elements.
problem Constructing an invariant for closed 3-manifolds.
method Using involutory Hopf algebra elements and normal o-graphs.
result Invariant values in cyclic quotient of Heisenberg double.
Develops a Gaussian model to compute the Alexander polynomial of knots.
problem Computing the Alexander polynomial of knots.
method Uses perturbed Gaussian functions, Heisenberg algebra, and tensor-contraction formalism.
result Associates a Gaussian function to a knot whose partition function recovers the Alexander polynomial.
We consider surfaces of class C1 in the 3-dimensional sub-Riemannian Heisenberg group H1. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In cas…
This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using KK-theory.
problem Analyzing the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators.
method Applying Kasparov's methodology and examining specific conditions using Fourier transform of the nilpotent group C∗-algebra. result Demonstrated enhanced methods for analyzing hypoellipticity and defined transversal Heisenberg ellipticity in a KK-theoretic context. This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, or Heisenberg calculus. The Heisenberg manifolds generalize CR and contact manifolds and in this context the main differential operators at stake include the Hörmander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian and its conf…
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.
Clarifies a trace for Heisenberg operators on contact manifolds.
problem Calculating the index of Heisenberg elliptic operators on contact manifolds.
method Introduced a new trace on Heisenberg pseudodifferential operators and constructed a cocycle in periodic cyclic cohomology.
result Simplified the construction of the trace on Heisenberg pseudodifferential operators.
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.
Extends Heisenberg homology to ribbon graphs.
problem Configurations in bounded surfaces.
method Regular thickening of ribbon graphs.
result Heisenberg homology applied to ribbon graphs.
In this paper we consider smooth oriented hypersurfaces in 2-step nilpotent Lie groups with a left invariant metric and derive an expression for the Laplacian of the Gauss map for such hypersurfaces in the general case and in some particular cases. In the case of CMC-hypersurface in the (2m+1)-dimensional Heisenberg gr…
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. 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 lifts plane mappings to Heisenberg group.
problem Contact quasiconformal mappings in hyperbolic Heisenberg group.
method Lifting Theorem for symplectic mappings.
result Symplectic mappings lifted to Heisenberg group.
We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as wel…
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.
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.
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.
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.
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…
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.
Compact metrics on Heisenberg manifolds have a specific condition for being relatively compact.
problem Conditions for relatively compact sets of left invariant metrics on Heisenberg manifolds.
method Necessary and sufficient condition for relatively compact sets of left invariant metrics.
result A condition for a set of left invariant metrics to be relatively compact in the moduli space.
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.
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.
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…
Study sub-Riemannian geodesics on a Heisenberg 3D nil-manifold.
problem Characterize geodesics on a specific nil-manifold.
method Analyze the projection of sub-Riemannian structure, describe geodesic flow dynamics, and estimate sub-Riemannian geodesics.
result Sharp bounds and estimates for sub-Riemannian geodesics.
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.
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.
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.
Study shows compact Sasakian manifolds are locally Heisenberg up to deformation.
problem Characterizing compact Sasakian manifolds.
method Analyzing basic Chern classes and using left invariant Sasakian structures.
result Compact Sasakian manifolds are locally isomorphic to the real Heisenberg group.
We study immersed, connected, umbilic hypersurfaces in the Heisenberg group Hn with n ≥ 2. We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigm…
Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.
problem Characterizing Sasakian manifolds with nilpotent fundamental groups.
method Proved diffeomorphism to Heisenberg nilmanifolds.
result Compact aspherical Sasakian manifolds with nilpotent fundamental groups are Heisenberg nilmanifolds.