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.
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48 results for “Rumin's differential forms”
New CR invariant treatment of Rumin complex via differential forms.
problem CR invariant treatment of Rumin complex.
method New treatment via differential forms, identification of balanced -structures, Hodge decomposition theorems.
result Sharp upper bound on Kohn--Rossi groups and CR analogue of Frölicher inequalities.
Derives properties of heat kernel for Rumin complex on Heisenberg groups.
problem Analyzing heat kernel properties for Rumin complex.
method Derives properties of heat equation with Hodge operator on Heisenberg groups.
result Constructs Calderón reproducing formula using heat kernel for Rumin forms.
Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.
problem Understanding magnetic flows on 3D contact sub-Riemannian manifolds.
method Introducing horizontal magnetic flows via closed Rumin differential two-forms and analyzing the lifted sub-Riemannian structure.
result Horizontal magnetic flows can be interpreted as geodesic flows on a suitably lifted structure, which is of Engel type when the magnetic field is non-vanishing.
Simple construction of Rumin algebra for contact manifolds.
problem Computing the de Rham cohomology algebra of contact manifolds.
method Using Markl's Homotopy Transfer Theorem for a simple explicit construction.
result Recovery of the Rumin algebra as a contact invariant -algebra.
We construct a versal family of deformations of CR structures in five dimensions, using a differential complex closely related to the differential form complex introduced by Rumin for contact manifolds.
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.
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.
Researchers prove inequalities for differential forms in Heisenberg groups, extending Euclidean results.
problem Quantitative formulations of topological problems in stratified Lie groups.
method Use of Rumin's complex and Poincaré/Sobolev inequalities for differential forms.
result Extension of -inequalities to Heisenberg groups for forms of degree at least 2.
Compact currents and charges in Carnot groups proved.
problem Compactness of normal currents in Carnot groups.
method Dual compactness argument for Rumin forms using pseudo-differential calculus.
result Compactness of normal currents in Carnot groups in flat topology.
The paper proves a decomposition theorem for forms on sub-Riemannian contact manifolds.
problem Developing a -Hodge decomposition on sub-Riemannian contact manifolds.
method Using a Sobolev approach and recent results from [4] and [6].
result Established an -Hodge decomposition theorem for Rumin's forms on sub-Riemannian contact manifolds.
Researchers compute determinants and torsions of Rumin complex in specific Lie group representations.
problem Computing determinants and torsions of Rumin complex in specific Lie group representations.
method Analyzing Schrodinger and generic representations of the (2,3,5) nilpotent Lie group.
result Computed the spectrum and zeta regularized determinant of Rumin differentials in Schrodinger representations and evaluated their alternating product in generic representations.
Harmonic forms and Rumin complex linked on Sasakian manifolds.
problem Relationship between harmonic forms and Rumin complex on Sasakian manifolds.
method Analytic torsion function and Rumin complex analysis.
result Kernel of Rumin Laplacian matches Hodge-de Rham Laplacian on compact Sasakian manifolds.
Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.
problem Understanding submanifolds with boundaries in Heisenberg groups.
method Introduce and study -regular submanifolds with boundaries, prove Stokes' Theorem for them.
result Prove a version of Stokes' Theorem for submanifolds with boundaries in Heisenberg groups.
The purpose of these notes is to explain parts of Gromov's survey of Carnot-Carathedory spaces, in the light of subsequent results of M. Rumin. Among the rich material provided by Gromov, most of which pertains to analysis on metric spaces, we choose to concentrate on the H{ö}lder equivalence problem for Carnot manifol…
Sobolev mappings preserve the Rumin complex on contact manifolds.
problem Preserving the Rumin complex under Sobolev mappings on contact manifolds.
method Using the Pullback Theorem, Pansu pullback is shown to induce chain mappings between Rumin complexes and de Rham complexes.
result The Rumin flat complex is bilipschitz invariant under Sobolev mappings between contact manifolds.
The paper compares three hypoelliptic Laplacians on a specific 5D Cartan group.
problem Defining suitable hypoelliptic Laplacians for sharp estimates on Carnot groups.
method Introducing and comparing three hypoelliptic Laplacians on a specific Carnot group.
result Sharp div-curl type inequalities for the three hypoelliptic Laplacians.
Let be a semisimple Lie group with finite center, a maximal compact subgroup, and a parabolic subgroup. Following ideas of P.Y.\ Gaillard, one may use -invariant differential forms on to construct -equivariant Poisson transforms mapping differential forms on to …
New definition of Rumin complex for nilpotent Lie groups.
problem No new problem introduced.
method Alternative definition of Rumin complex on nilpotent Lie groups.
result Direct application of cohomology results to all nilpotent Lie groups.
In this paper, we prove interior Poincar{é} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integ…
Eta invariant of (2,3,5) nilmanifolds vanishes but eta function is nontrivial.
problem Eta invariant of (2,3,5) nilmanifolds.
method Study of eta function and invariant of a self-adjoint differential operator of Heisenberg order two.
result Formula expressing the eta function of (2,3,5) nilmanifolds in terms of elementary functions.
The paper explores the Rumin complex and spectral sequence on Carnot groups.
problem Understanding the relationship between Rumin complex and spectral sequence on Carnot groups.
method Investigates the Rumin complex and spectral sequence on Carnot groups, focusing on the filtration by homogeneous weights.
result Provides a detailed insight into the relationship between the Rumin complex and the spectral sequence on Carnot groups.
We propose a definition for analytic torsion of the Rumin complex on contact manifolds. This is given by the derivative at zero of a well-chosen combination of zeta functions of a fourth-order modified Rumin Laplacian. The regular value at zero (before differentiation) of this well-chosen combination of zeta functions …
Paper proves inequalities for forms on sub-Riemannian manifolds.
problem Establishing inequalities for differential forms on sub-Riemannian contact manifolds.
method Using structure of Rumin's complex, Sobolev-Gaffney inequality for Heisenberg groups, and geometric properties.
result Gaffney type inequality in Sobolev spaces for differential forms on sub-Riemannian contact manifolds with bounded geometry.
Given a contact manifold $M_#$ together with a transversal infinitesimal automorphism , we show that any local leaf space for the foliation determined by naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on $M_#$ descends to a complex of differential operators on…
This paper constructs Poisson transforms and analyzes their properties on complex hyperbolic spaces.
problem Understanding discrete series representations of SU(n+1,1) using differential forms.
method Constructing Poisson transforms and analyzing their boundary asymptotics and intertwining properties with the Rumin complex.
result The constructed transforms realize the direct sum of all discrete series representations of SU(n+1,1).
This paper shows that when the Riemannian metric on a contact manifold is blown up along the direction orthogonal to the contact distribution, the corresponding harmonic forms rescaled and normalized in the -norms will converge to Rumin's harmonic forms. This proves a conjecture in Gromov `` Carnot-Caratheodory sp…
Analytic torsions on contact spheres are calculated using Rumin complex.
problem Calculating analytic torsions for contact spheres.
method Explicitly wrote down eigenvalues of Rumin Laplacian and expressed analytic torsion functions in terms of Riemann zeta function.
result Functions of analytic torsions vanish at the origin and were determined.
The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.
problem Validating Stokes' theorem for differential subcomplexes in positively graded Lie groups.
method Introducing geometric conditions and spectral complexes to recover Stokes' theorem on locally smooth intrinsic graphs.
result Stokes' theorem holds for Rumin complex and new spectral complexes on Carnot groups.
Alternative construction of Rumin complex on Lie groups.
problem Constructing Rumin complex on homogeneous nilpotent Lie groups.
method Using ideas from parabolic geometry, an alternative construction to the classical one on Carnot groups.
result Explicit computations for the Engel group using the new approach.
Explicitly expresses torsion functions on lens spaces.
problem Calculating torsion on lens spaces.
method Expressed torsion functions in terms of Hurwitz zeta function and Ray-Singer torsion.
result Found that torsion functions vanish at the origin and determined their values.
New complexes refine multicomplexes for subRiemannian geometry.
problem Analyzing subRiemannian geometry on Carnot groups.
method Spectral complexes from truncated multicomplexes.
result Retains cohomology of multicomplexes and refines Rumin complex.
The paper proves Rademacher's theorem for Heisenberg groups.
problem Proving Rademacher's theorem for Heisenberg groups.
method New definition of intrinsic Lipschitz graphs, extension and approximation theorems, use of Heisenberg currents and Rumin's complex.
result Rademacher's theorem for intrinsic Lipschitz graphs in Heisenberg groups.
This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features such as trajectories and length. We also define several differential operators on q…
Spectral sequence analysis for Sobolev mappings in Carnot groups.
problem Analyzing spectral sequences for Sobolev mappings in Carnot groups.
method Showed Pansu pullback induces a spectral sequence mapping.
result Pansu pullback induces a spectral sequence mapping.
This paper defines and studies currents and slices in the Heisenberg group, with new challenges and insights.
problem Defining and studying currents and slices in the Heisenberg group .
method Definition and classification of currents, slicing of currents, and analysis of properties.
result New challenges and insights in the study of currents on the Heisenberg group, including a unique slice dimension.
The Alesker-Poincare pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pai…
In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric projections. In the CR setting these operators arise from the Kohn-Rossi complex and include the Szegö projections on forms. In the contact setting they stem from the general…
Analytic torsion defined for rank 2 distributions on 5-manifolds.
problem Defining and analyzing analytic torsion for rank 2 distributions.
method Proposed an analytic torsion for Rumin complex associated with rank 2 distributions on 5-manifolds, established anomaly formulas, and showed coincidence with Ray-Singer torsion.
result The proposed torsion coincides with Ray-Singer torsion for certain nilmanifolds.
Paper constructs subcomplexes from filtered Riemannian manifolds.
problem Understanding subcomplexes on filtered Riemannian manifolds.
method General construction of subcomplexes from two distinct complexes.
result Aligns with Rumin complex on regular subRiemannian manifolds.
Analytic torsion matches Ray-Singer for specific nilmanifolds.
problem Matching analytic torsion with Ray-Singer in specific nilmanifolds.
method Examined rank two distributions on 5D nilmanifolds, proving torsion equality.
result Analytic torsion equals Ray-Singer torsion in these specific nilmanifolds.
The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper we establish a universa…
GROVER improves word representations by gradually adding random noises during training.
problem Improving word representations for better model performance.
method Gradually adding random noises to word embeddings during training.
result GROVER improves model performances on most text classification datasets.
We relate -cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of -cohomology, packing cohomology. This implies quasi-isometry invariance of -cohomology together with its multiplicative structure. The result partially extends to the Rumin -cohomolog…
Criterion for lifting smooth contact maps between Carnot groups to central extensions.
problem Existence of smooth contact map lifts between Carnot groups and their central extensions.
method Criterion using pullbacks and Lie algebra cohomology classes.
result Necessary and sufficient conditions for lifting are formulated.
The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures. The…
Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
New Poincaré inequality for differential forms on manifolds.
problem Developing inequalities for differential forms on manifolds.
method Proving a new Poincaré-type inequality and deriving new inequalities involving mean and scalar curvatures.
result Characterized the limiting case of a new inequality involving mean and scalar curvatures of the boundary.