Lefschetz pencil structures for 2-calibrated manifolds

We prove that for closed 2-calibrated manifolds there always exist Lefschetz pencil structures. This generalizes similar results for symplectic and contact manifolds. To cite this article: A. Ibort, D. Marti´nez Torres, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Sur la continuité de Hölder du semi-groupe de la chaleur sur les variétés coniques

In this Note, we study initially the heat kernel, p_t, on conic manifolds of dimension 2. Then, we improve the upper bound of p_t obtained in Li (Bull. Sci. Math. 124 (2000) 365–384) on conic manifolds of dimension ?3. Finally, we study the Hölder continuity of the heat semigroup on conic manifolds. Some new phenomenons are found on conic manifolds, in particular, on conic manifolds of dimension 2. To cite this article: H.-Q. Li, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A sharp Sobolev inequality on Riemannian manifoldsUne inégalité de Sobolev optimale sur les varietés riemanniennes

We outline our results in [11] concerning some sharp Sobolev inequalities on Riemannian manifolds. Our inequalities emphasize the role of scalar curvature in this context. To cite this article: Y.Y. Li, T. Ricciardi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 519–524.     

Sur les variétés sous-riemanniennes de contact isotropes

In this Note, we show that contrary to the dimension 3 case, isotropic contact sub-Riemannian manifolds of dimension greater than 3 do not exist. To cite this article: A.-R. Mansouri, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A stochastic differential equation from friction mechanics

The existence and uniqueness of solutions to multivalued stochastic differential equations of the second order on Riemannian manifolds are proved. The class of problem is motivated by rigid body and multibody dynamics with friction and an application to the spherical pendulum with friction is presented. To cite this article: F. Bernardin et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Comparaison des volumes des variétés riemanniennes

Using the rigidity result of Besson, Courtois and Gallot, and also the notion of intersection of metrics, we compare volumes of Riemannian manifolds by means of lengths of their periodic geodesics. To cite this article: H.-R. Fanaï, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Symplectic capacities of toric manifolds and combinatorial inequalitiesCapacités symplectiques de variétés toriques et inéqualités combinatoires

We shall give concrete estimations for the Gromov symplectic width of toric manifolds in combinatorial data. As by-products some combinatorial inequalities in the polytope theory are obtained. To cite this article: G. Lu, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 889–892.     

The Ω Deformed B-model for Rigid N = 2 Theories

We give an interpretation of the Ω deformed B-model that leads naturally to the generalized holomorphic anomaly equations. Direct integration of the latter calculates topological amplitudes of four-dimensional rigid N = 2 theories explicitly in general Ω-backgrounds in terms of modular forms. These amplitudes encode the refined BPS spectrum as well as new gravitational couplings in the effective action of N = 2 supersymmetric theories. The rigid N = 2 field theories we focus on are the conformal rank one N = 2 Seiberg–Witten theories. The failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation. Our formalism applies irrespectively of whether a Lagrangian formulation exists. In the class of rigid N = 2 theories arising from compactifications on local Calabi–Yau manifolds, we consider the theory of local ${\mathbb{P}^2}$ . We calculate motivic Donaldson–Thomas invariants for this geometry and make predictions for generalized Gromov–Witten invariants at the orbifold point.     

Generalized scattering phases for asymptotically hyperbolic manifolds

We prove asymptotic expansions of generalized scattering phases asssociated to pairs of Laplacians, for a class of noncompact manifolds with infinite volume and negative curvature near infinity. We use one of these expansions to define relative determinants which appear naturally in this context. To cite this article: J.-M. Bouclet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sharp constants in weighted trace inequalities on Riemannian manifolds

In this paper, we establish some sharp weighted trace inequalities ${W^{1,2}(\rho^{1-2 \sigma}, M) \hookrightarrow L^{\frac{2n}{n-2 \sigma}}(\partial M)}$ on n + 1 dimensional compact smooth manifolds with smooth boundaries, where ρ is a defining function of M and ${\sigma \in (0,1)}$ . This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.     

A spectra comparison theorem and its applications

We give a sharp comparison between the spectra of two Riemannian manifolds (Y, g) and \((X,g_0)\) under the following assumptions: \((X,g_0)\) has bounded geometry, (Y, g) admits a continuous Gromov–Hausdorff \(\varepsilon \)-approximation onto \((X,g_0)\) of non zero absolute degree, and the volume of (Y, g) is almost smaller than the volume of \((X,g_0)\). These assumptions imply no restrictions on the local topology or geometry of (Y, g) in particular no curvature assumption is supposed or inferred.     

The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation

We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. A Riemannian interpretation of the Reeb class will give some geometric criteria which enables one to tell whether the modular class vanishes or not. It also enables one to construct examples of unimodular Poisson manifolds and others which are not unimodular. Finally, we prove that the first leafwise cohomology space is an invariant of Morita equivalence. To cite this article: A. Abouqateb, M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Équations différentielles stochastiques conduites par des lacets dans les groupes de Carnot

The subriemannian geometry of stochastic differential equations driven by processes generating loops in free Carnot groups are studied. To cite this article: F. Baudoin, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The symplectic reduced spaces of a Poisson actionLes espaces réduits symplectiques d'une action de Poisson

During the last thirty years, symplectic or Marsden–Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the presence of a momentum map. In this Note we show that the symplectic reduction phenomenon has much deeper roots. More specifically, we will find symplectically reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. In other words, the right category to obtain symplectically reduced spaces is that of Poisson manifolds acted upon canonically by a Lie group. To cite this article: J.-P. Ortega, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 999–1004.     

Propriétés génériques des trajectoires singulières

We give genericity results for singular trajectories in sub-Riemannian geometry: generically (in the sense of the Whitney topology), every singular trajectory is of minimal order and of corank 1 and in particular is not of Goh type if the rank of the distribution is greater or equal to 3. We extend these results to control-affine systems. To cite this article: Y. Chitour et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Volume-convergent sequences of Haken 3-manifolds

Let M be a closed orientable 3-manifold and let Vol(M) denote its Gromov simplicial volume. This paper is devoted to the study of sequences of non-zero degree maps $\text{f}{}_{\mathrm{i}}\text{:M→N}{}_{\mathrm{i}}$ to Haken manifolds. We prove that any sequence of Haken manifolds (N_i,f_i), satisfying lim_i→∞deg(f_i)×Vol(N_i)=Vol(M) is finite up to homeomorphism. As an application, we deduce from this fact that any closed orientable 3-manifold with zero Gromov simplicial volume and in particular any graph manifold dominates at most finitely many Haken 3-manifolds. To cite this article: P. Derbez, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Variétés complexes dont l'éclatée en un point est de FanoComplex manifolds whose blow-up at a point is Fano

We classify complex projective manifolds X for which there exists a point a such that the blow-up of X at a is Fano. As a consequence, we get that, in dimension greater or equal than three, the quadric is the only complex manifold X for which there exists two distinct points a and b such that the blow-up of X with center {a,b} is Fano. To cite this article: L. Bonavero et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 463–468.     

A reduction of the slicing problem to finite volume ratio bodies

Here we discuss results around the slicing problem, which is a well known open problem in asymptotic convex geometry. We show that if one can prove that the isotropic constant of bodies with a finite volume ratio is uniformly bounded – then it would follow that the isotropic constant of any convex body is uniformly bounded. To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A generalization of K-contact and (k, μ)-contact manifolds

We study a generalization of K-contact and (k, μ)-contact manifolds, and show that if such manifolds of dimensions ≥ 5 are conformally flat, then they have constant curvature +1. We also show under certain conditions that such manifolds admitting a non-homothetic closed conformal vector field are isometric to a unit sphere. Finally, we show that such manifolds with parallel Ricci tensor are either Einstein, or of zero ${\xi}$ -sectional curvature.     

Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles

The pressure function P(A, s) plays a fundamental role in the calculation of the dimension of “typical” self-affine sets, where A = (A ₁, …,A _k ) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A. As a consequence, we show that the dimension of “typical” self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.