arXiv research

Extends De Leeuw theorems to noncommutative groups and multipliers.

Local norms of Fourier multipliers bounded on discrete subgroups of Lie groups.

The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.

Paper extends positive energy theorem to anti-de Sitter spacetimes.

Study of de Rham cohomology on non-Hausdorff manifolds.

New Lipschitz de Rham theorem for $L_p$-cohomology.

The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.

Paper proves rigidity for certain spacelike hypersurfaces in de Sitter space.

The Burde--de Rham theorem is extended to finitely presented pro-$p$ groups with specific conditions.

De Rham theorem extended to Orlicz cohomology.

We prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.

Proof of Thurston's earthquake theorem using Anti-de Sitter geometry.

Generalizes Candel's theorem on curvature of laminated surfaces.

Study examines solutions to Jang equation on anti-de Sitter spacetimes.

We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.

We define the total energy-momenta for (4+1)-dimensional asymptotically anti-de Sitter spacetimes, and prove the positive energy theorem for such spacetimes.

Uniqueness theorem for extremal charged black holes in de Sitter space.

By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisify a stratified version of Whitehead's theorem. As an example, we introd…

This paper proves a positive energy-momentum theorem for oriented Riemannian 3-manifolds that are asymptotic to a standard hyperbolic slice in anti de Sitter space-time. Analogously to the original Witten's proof in the asymptotically flat case, this result relies on spinorial methods. We also give a rigidity theorem: …

We prove an analogue of the de Rham theorem for the extended L^2-cohomology introduced by M. Farber. This is done by establishing that the de Rham complex over a compact closed manifold with coefficients in a flat Hilbert bundle E of A-modules over a finite von Neumann algebra A is chain-homotopy equivalent (with bound…

Promotes spectral functionals to noncommutative fields and proves a theorem.

We show that the de Rham theorem, interpreted as the isomorphism between distributional de Rham cohomology and simplicial homology in the dual dimension for a simplicial decomposition of a compact oriented manifold, is a straightforward consequence of elementary properties of currents. The explicit construction of this…

The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a nove…

We use planar coordinates as well as hyperbolic coordinates to separate the de Sitter spacetime into two parts. These two ways of cutting the de Sitter give rise to two different spatial infinities. For spacetimes which are asymptotic to either half of the de Sitter spacetime, we are able to provide definitions of the …

Explains Hodge theory and Kodaira embedding theorem for complex manifolds.

We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We…

Study on constant mean curvature hypersurfaces in Anti-de Sitter space.

Geometric transformations on null curves in AdS induce KdV solutions.

This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes' theorem, De Rham cohomology, Poincare duality, and integration along the fiber. The resulting cohomology theory is used to define Gromov Witten invariants of exploded manifolds in a separate paper.

Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be…

Study the structure of Kähler foliations with negative Ricci curvature.

Dans les années 1940-1970, Alexandrov et l'"École de Leningrad" ont développé une théorie très riche des surfaces singulières. Il s'agit de surfaces topologiques, munie d'une métrique intrinsèque pour laquelle on peut définir une notion de courbure, qui est une mesure de Radon. Cette classe de surfaces a de bonnes prop…

In this paper, we extend the classical de Rham decomposition theorem to the case of Riemannian manifolds with boundary by using the trick of development of curves.

A new proof of Friedrich's theorem on the existence and stability of asymptotically de Sitter spaces in 3+1 dimensions is given, which extends to all even dimensions. In addition, we characterize the possible limits of spaces which are globally asymptotically de Sitter, to the past and future.

We generalize a theorem of Burde and de Rham characterizing the zeros of the Alexander polynomial. Given a representation of a knot group $π$, we define an extension of $π$, the Crowell group. For any GL(n,C) representation of $π$, the zeros of the associated twisted Alexander polynomial correspond to representations o…

We establish the inequality for Henneaux-Teitelboim's total energy-momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary $t$-slice in anti-de Sitter spacetime. In particular, when $t=0$, it generalizes Chruściel-Maerten-Tod's inequality in the center of AdS mass coordinates. We …

Extends Hopf's theorem to de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.

Simple construction of Rumin algebra for contact manifolds.

De Sitter spacetime can be separated into two parts along two kinds of hypersurfaces and the half-de Sitter spacetimes are covered by the planar and hyperbolic coordinates respectively. Two positive energy theorems were proved previously for certain $¶$-asymptotically de Sitter and $\H$-asymptotically de Sitter initial…

On the slit tangent manifold of a Finsler manifold M are given the vertical and the Liouville foliations. In this paper we define some new types of vertical forms with respect to the Liouville foliation on TM^0. We define a cohomology group of TM^0 using these new forms. We prove a de Rham type theorem.

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds (of which symplectic manifolds are an important class of examples). Quantum de Rham cohomology, which is a deformation quantization of de Rham cohomology, is defined as the cohomology of d_h. We also define quantum Dol…

The paper extends Hodge-de Rham and Lichnérowicz Laplacians to double forms and proves vanishing theorems.

We endow the de Rham cohomology of any Poisson or Jacobi manifold with a natural homotopy Frobenius manifold structure. This result relies on a minimal model theorem for multicomplexes and a new kind of a Hodge degeneration condition.

We consider a compact connected CR manifold with a transversal CR locally free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion and we establish $\mathbb R$-equivariant K…

We establish a type of positive energy theorem for asymptotically anti-de Sitter Einstein-Maxwell initial data sets by using Witten's spinoral techniques.

This article is based on the lectures given at the ``Ecole thematique de theorie ergodique'', at the C.I.R.M. in Marseille, in April 2006. We give a complete proof of a theorem of Kerckhoff, Masur and Smillie on the unique ergodicity of the directional flow on a translation surface in almost every direction. The proof …

New cohomology theory for diffeological spaces developed.

Hopf's theorem generalized to curved spaces.