arXiv research

Sharp stability of isometries on Heisenberg group proven.

In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We also prove a comparison principle for solutions of second order fully…

Study on surfaces in Heisenberg group with constant mean curvature.

This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of Heisenberg PsiDOs introduced by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order Heisenberg PsiDOs…

In Heisenberg groups, rectifiability is studied for subsets using $C^{1,α}$-regular surfaces.

We study the horizontally regular curves in the Heisenberg groups $H_n$. We show the fundamental theorem of curves in $H_n$ $(n\geq 2)$ and define the concept of the orders for horizontally regular curves. We also show that the curve $γ$ is of order $k$ if and only if $γ$ lies in $H_k$ but not in $H_{k-1}$ up to a Heis…

It is a folk conjecture that for alpha > 1/2 there is no alpha-Hoelder surface in the subRiemannian Heisenberg group. Namely, it is expected that there is no embedding from an open subset of R^2 into the Heisenberg group that is Hoelder continuous of order strictly greater than 1/2. The Heisenberg group here is equippe…

Maps in Carnot groups are equivalent to solutions of a PDE system.

We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step nilpotent groups. Some consequences of this work are a construction of groups with ar…

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result i…

We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this p…

Energy functional for Legendrian knots in Heisenberg group, invariant under PU(2,1).

We study in this paper, the existence of tree types of filtrations of the space $\mathcal{D}_{λμ}(\R^{2l+1|n})$ of differential operators on the superspaces $\R^{2l+1|n}$ endowed with the standard contact structure $α$. On this space $\mathcal{D}_{λμ}(\R^{2l+1|n})$, we have the first filtration called canonical and bec…

Eta invariant of (2,3,5) nilmanifolds vanishes but eta function is nontrivial.

We establish a "low rank property" for Sobolev mappings that pointwise solve a first order nonlinear system of PDEs, whose smooth solutions have the so-called "contact property". As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group and that have almost everywhere…

This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using $KK$-theory.

New smooth solutions of the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton based on the quaternionic Heisenberg group are constructed. We show that through appropriate contractions the solutions found in the $G_2$-heterotic case converge to the heterotic solutions on 6-dimensi…

It is known that the spectrum of the Laplace operator on functions of a closed Riemannian manifold does not determine the integrals of the individual fourth order curvature invariants $\operatorname{scal}^2$, $|\operatorname{ric}|^2$, $|R|^2$, which appear as summands in the second heat invariant $a_2$. We study the an…

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.

Clarifies a trace for Heisenberg operators on contact manifolds.

Homogeneous magnetic paths found in Heisenberg space.

Extends Heisenberg homology to ribbon graphs.

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

Classifies geodetically convex sets and functions on Heisenberg group.

New conformally Einstein metrics on Heisenberg group found.

We construct a non-formal deformation machinery for the actions of the Heisenberg supergroup analogue to the one developed by M. Rieffel for the actions of R^d. However, the method used here differs from Rieffel's one: we obtain a Universal Deformation Formula for the actions of R^{m|n} as a byproduct of Weyl ordered K…

New knot invariants derived using quantum cluster algebras.

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.

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.

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…

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

Compact metrics on Heisenberg manifolds have a specific condition for being relatively compact.

Study shows only hyperplanes in Heisenberg groups have zero curvature.

Researchers prove inequalities for differential forms in Heisenberg groups, extending Euclidean results.

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

Translation surfaces in Heisenberg group classified by Gauss map determinant.

Study maps surface configurations to Heisenberg homologies for mapping class 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.

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…

Researchers define geodesic curvature for 3D sub-Riemannian curves, improving distance calculations.

Study sub-Riemannian geodesics on a Heisenberg 3D nil-manifold.