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.
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…
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.
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.
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.
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.
Paper shows limits of Heisenberg manifolds are flat tori.
problem Understanding limits of sub-Riemannian Heisenberg manifolds.
method Analyzes collapsed Gromov--Hausdorff limits of compact Heisenberg manifolds.
result Collapsed limits are isometric to flat tori.
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.
As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold M. As it is well known for a Heisenberg manifold (M,H) the relevant notion of tangent is…
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.
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.
This note describes a canonical way to orient the Heisenberg 3-manifold.
The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.
problem The study of limits of sub-Riemannian Heisenberg manifolds and their properties.
method Analysis of Gromov-Hausdorff limits with upper and lower bounds on diameter and Popp's measure.
result The non-collapsed limits of sub-Riemannian Heisenberg manifolds converge to a compact Heisenberg manifold.
Classifies Heisenberg-invariant self-dual Einstein manifolds with explicit metrics.
problem Classifying self-dual Einstein manifolds invariant under Heisenberg group actions.
method Explicit construction of metrics and analysis of completeness.
result Einstein constants can vary and solutions exist for non-zero Ricci curvature.
In this paper, we study the Yang-Mills functional on quantum Heisenberg manifolds using the appratuses developed by A. Connes and M. Rieffel. It is discovered that a connection on a projective module over a quantum Heisenberg manifold is a minimum of Yang-Mills functional whicih is a critical point that is different wi…
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…
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. 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 magnetic geodesics on Heisenberg groups and manifolds.
problem Dynamics of magnetic flows on Heisenberg groups.
method Explicit description of magnetic geodesics, determination of lengths.
result Density of periodic magnetic geodesics and marked magnetic length spectrum rigidity.
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.
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…
We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher…
Researchers construct an index map for contact manifolds using K-theory.
problem Constructing an index for maximally hypoelliptic operators on contact manifolds.
method Using Higson's construction for symbol class in K-theory, they derive a series of maps whose induced map in K-theory is the Heisenberg Atiyah-Singer index map.
result Explicit construction of a series of maps leading to the Heisenberg Atiyah-Singer index map.
The paper extends Pappus-Guldin theorems to 3D-Heisenberg group surfaces.
problem Extending classical theorems to a new geometric setting.
method Deriving formulas for p-areas and volumes in the Heisenberg group.
result Pappus-Guldin theorems hold for surfaces in the Heisenberg group.
One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental inte…
Proves inequalities on curved spaces with positive curvature.
problem Proving inequalities on manifolds with nonnegative Ricci curvature.
method Analyzes manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Proves Heisenberg-Pauli-Weyl, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities.
Classifies Einstein metrics on R^4 with Heisenberg symmetry, finding incomplete Ricci-flat metrics and two complete negative-curvature examples.
problem Classifying Einstein metrics on R4 with Heisenberg symmetry. method Invariant under a four-dimensional group of isometries including the Heisenberg group, analyzing Ricci-flat and negative-curvature metrics.
result Found two complete negative-curvature examples: complex hyperbolic metric and one-loop deformed universal hypermultiplet.
New proof of wave trace formula for 3D-contact manifolds.
problem Wave trace formula for 3D-contact manifolds.
method Normal form reduction to Heisenberg group.
result Extension of Chazarain-Duistermaat-Guillemin formula.
Defines Wodzicki residue using groupoids and fibered distributions.
problem Defining and understanding the Wodzicki residue in noncommutative geometry.
method Using groupoid language and filtered manifolds, defining the residue and showing its properties.
result The groupoidal residue is a trace on pseudodifferential operators and matches the usual residue in certain cases.
A complete solution to the quaternionic contact Yamabe equation on the qc sphere of dimension 4n+3 as well as on the quaternionic Heisenberg group is given. A uniqueness theorem for the qc Yamabe problem in a compact locally 3-Sasakian manifold is shown.
The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single e…
This paper extends IMCF theory to Heisenberg group, solving Penrose inequality.
problem Extending inverse mean curvature flow theory to Heisenberg group.
method Developed a sub-Riemannian theory for Heisenberg group, introduced a flow preserving H-perimeter.
result Established a Minkowski-type formula in Heisenberg group, proving Heintze-Karcher inequality.
Zero-energy orbits in the Kepler-Heisenberg problem are self-similar and stratify into three families.
problem Determining the motion of a planet around a sun in the Heisenberg group.
method Analysis of the sub-Riemannian Hamiltonian and sub-Laplacian dynamics.
result Zero-energy orbits are self-similar and stratify into future collision, past collision, and quasi-periodic families.
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.
This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg …
Poincar{é} and Sobolev inequalities for differential forms on Heisenberg balls, involving Rumin's differentials, are given. Furthermore, a global homotopy of Rumin's complex which improves differentiability of Rumin forms is provided on any bounded geometry contact manifold.
New method constructs degenerate Sasakian manifolds from hyperkähler bundles.
problem Constructing degenerate 3-(α,δ)-Sasakian manifolds. method Using fiber products of Boothby-Wang bundles over hyperkähler manifolds.
result No non-trivial compact examples exist, and one family of nilpotent Lie groups with this geometry is identified.
Characterizes polyhomogeneous symbols and applies to Heisenberg calculus.
problem Understanding polyhomogeneous symbols and their applications.
method Simple characterisation and generalization of A.~Connes' tangent groupoid.
result Heisenberg calculus on contact manifolds coincides with groupoid calculus.
Injective X-ray transform on Heisenberg group for regular functions.
problem Injectivity of X-ray transform on sub-Riemannian manifolds.
method Group Fourier Transform and analysis of taming metrics.
result Sufficiently regular functions on Heisenberg group are determined by their line integrals.
A contact manifold is a manifold equipped with a distribution of codimension one that satisfies a `maximal non-integrability' condition. A standard example of a contact structure is a strictly pseudoconvex CR manifold, and operators of analytic interest are the tangential Cauchy-Riemann operator and the Szego projector…
New steady Euler flows found on 3-sphere and Sasakian manifolds.
problem Finding new steady Euler solutions on specific manifolds.
method Bifurcating from an existing ansatz and extending it to Sasakian 3-manifolds.
result Previously known solutions are not isolated, and new solutions found on 3-sphere and other Sasakian manifolds.
We consider the biharmonicity condition for maps between Riemannian manifolds (see [BK]), and study the non-geodesic biharmonic curves in the Heisenberg group H_3. First we prove that all of them are helices, and then we obtain explicitly their parametric equations.
New star-product defined on Poisson manifolds using Toeplitz operators.
problem Defining star-products on Poisson manifolds induced by symplectic Lie algebroids.
method Using Toeplitz operators on groupoids with Heisenberg group structure.
result Generalization of Guillemin and Melrose's symplectic approach.
Study rigidity on CR Yamabe equation on Sasakian manifolds.
problem Proving rigidity on CR Yamabe equation on Sasakian manifolds.
method Using Jerison-Lee's differential identity and integral estimates.
result Prove that the manifold is CR isometric to Heisenberg group \(\mathbb{H}^n\).
A characterization of maximal domains of existence of adapted complex structures for Riemannian homogeneous manifolds under certain extensibility assumptions on their geodesic flow is given. This is applied to generalized Heisenberg groups and naturally reductive Riemannian homogeneous spaces. As an application it is s…
We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typ…
In infinite dimensional Heisenberg group, degenerate distances linked to unbounded curvature.
problem Degenerate distances and unbounded curvature in infinite dimensional Heisenberg group.
method Construct left invariant weak Riemannian and sub-Riemannian metrics, adapt sectional curvature definition.
result Degenerate distances coincide with unbounded sectional curvature.
The paper confirms a specific type of Sasakian manifold's structure.
problem Characterizing Sasakian manifolds with nonnegative transverse bisectional curvature.
method Analyzing the Sasakian analogue of Yau's uniformization conjecture.
result 5-dimensional Sasakian manifolds with positive transverse bisectional curvature are CR-biholomorphic to the standard Heisenberg group.