Publications

2007

• On the capacity of Lagrangians in the cotangent disc bundle of the torus
  
  • Viterbo Claude
  , 2007. This paper was withdrawn due to a critical error. For recent results in this direction, see Shelukhin's papers on the subject https://arxiv.org/abs/1904.06798 , https://arxiv.org/abs/1811.05552 and BIran-Cornea https://arxiv.org/abs/2008.04756 -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Former abstract: We prove that a Lagrangian torus in $T^*T^n$ Hamiltonianly isotopic to the zero section and contained in the unit disc bundle has bounded $\gamma$-capacity, where $\gamma(L)$ is the norm on Lagrangian submanifold. On one hand this gives new obstructions to Lagrangian embeddings of a quantitative kind. On the other hand, it gives a certain control on the $\gamma$ topology in terms f the Hausdorff topology. Finally this result is a crucial ingredient in establishing symplectic homogenization theory.
• Equivariant relative Thom forms and Chern characters
  
  • Paradan Paul-Emile
  • Vergne Michèle
  , 2007. These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomology. These relations include the Thom isomorphism in equivariant cohomology, the multiplicativity of the relative Chern characters, and the Riemann-Roch relation between the relative Chern character of the Bott symbol and of the relative Thom class.
• The Dirac spectrum on manifolds with gradient conformal vector fields
  
  • Moroianu Andrei
  • Moroianu Sergiu
  Journal of Functional Analysis, Elsevier, 2007, 253 (1), pp.207-219. We show that the Dirac operator on a spin manifold does not admit $L^2$ eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing. (10.1016/j.jfa.2007.04.013)
  DOI : 10.1016/j.jfa.2007.04.013
• Killing vector fields with twistor derivative
  
  • Moroianu Andrei
  Journal of Differential Geometry, International Press, 2007, 77 (1), pp.149-167. Motivated by the possible characterization of Sasakian manifolds in terms of twistor forms, we give the complete classification of compact Riemannian manifolds carrying a Killing vector field whose covariant derivative (viewed as a 2-form) is a twistor form.
• Radiative transfer equations and Rosseland approximation in gray matter
  
  • Salvarani Francesco
  • Golse François
  , 2008, pp.321-326. In this paper we consider the radiative transfer equations in a bounded domain with non-homogeneous boundary conditions, when the opacity does not depend on the frequency of the photons. We discuss the existence of weak solutions of the radiative transfer system and show that the corresponding Rosseland approximation is robust even when the target equation is parabolic degenerate and the flux on the boundary is non vanishing. (10.1142/9789812772350_0045)
  DOI : 10.1142/9789812772350_0045
• Convergence of Harder-Narasimhan polygons
  
  • Chen Huayi
  , 2007. We establish in this article convergence results of normalized Harder-Narasimhan polygons both in geometric and in arithmetic frameworks by introducing the Harder-Narasimhan filtration indexed by $\mathbb R$ and the associated Borel probability measure.
• On the cost of fast controls for thermoelastic plates
  
  • Miller Luc
  Asymptotic Analysis, IOS Press, 2007, 51 (2), pp.93--100. This paper proves that any initial condition in the energy space for the system of thermoelastic plates without rotatory inertia on a smooth bounded domain with hinged mechanical boundary conditions and Dirichlet thermal boundary condition can be steered to zero by a square integrable input function, either mechanical or thermal, supported in arbitrarily small sub-domain and time interval [0,T]. As T tends to zero, for initial states with unit energy norm, the norm of this input function grows at most like exp( C_p / T^p ) for any real p > 1 and some C_p > 0. These results are analogous to the optimal ones known for the heat flow and the proof uses the heat control strategy of Lebeau and Robbiano.
• Maximal Entropy Measures for Piecewise Affine Surface Homeomorphisms
  
  • Buzzi Jerome
  , 2007. We study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected ``good'' returns to dispense with Markov partitions. We control these good returns through some entropy and ergodic arguments.
• Infinitesimal Einstein Deformations of Nearly Kähler Metrics
  
  • Moroianu Andrei
  • Semmelmann Uwe
  , 2007. It is well-known that every 6-dimensional strictly nearly Kähler manifold $(M,g,J)$ is Einstein with positive scalar curvature $scal>0$. Moreover, one can show that the space $E$ of co-closed primitive (1,1)-forms on $M$ is stable under the Laplace operator $\Delta$. Let $E(a)$ denote the $a$-eigenspace of the restriction of $\Delta$ to $E$. If $M$ is compact, we prove that the moduli space of infinitesimal Einstein deformations of the nearly Kähler metric $g$ is naturally isomorphic to the direct sum $E(scal/15)\oplus E(scal/5)\oplus E(2scal/5)$. It is known that the last summand is itself isomorphic with the moduli space of infinitesimal nearly Kähler deformations.
• Autour des déformations de Rankin-Cohen.
  
  • Yao Yi-Jun
  , 2007. Dans cette thèse on s'attache à étudier les crochets de Rankin-Cohen et les déformations correspondantes selon de différents points de vue. On présente d'un côté une nouvelle interprétation des déformations de Rankin-Cohen via la théorie de "Quantification par Deformations de Fedosov(en collaboration avec P. Bieliavsky et X. Tang). On parvient notamment à redémontrer un théorème de Connes-Moscovici sur la déformation formelle des algèbres sous l'action d'une algèbre de Hopf H1 munie d'une structure projective. De l'autre cote on donne dans Chapitre III une interprétation détaillée des crochets de Rankin-Cohen via la théorie de représentations unitaires de SL2(R) et en utilisant cette interprétation on étudie certaines propriétés des produits déformés, notamment l'unicité des produits construits par Cohen-Manin-Zagier et une propriété de séparation du produit d'Eholzer. Dans le dernier chapitre on donne une démonstration élémentaire de l'identité combinatoire qui est cruciale pour démontrer l'associativité dans l'approche de la question de déformations par Cohen-Manin-Zagier, Eholzer, et Connes-Moscovici.
• Three theorems on perturbed KdV
  
  • Kuksin Sergei
  , 2008, pp.85-92. This short paper is based on a lecture, given at the NATO Advanced Study Institute on Hamiltonian dynamical systems (Montréal, 2007). Its goal is to discuss three theorems on the long-time behaviour of solutions of a perturbed KdV equation under periodic boundary conditions. These theorems are infinite-dimensional analogies of three classical results on small perturbations of an integrable finite dimensional system: - The KAM theorem - The first-order averaging theory for Hamiltonian perturbations - The Khasminskii averaging theory for random perturbations The three theorems raise many new questions, some of which are mentioned below. We stress that the three theorems are infinite-dimensional analogies of some finite-dimensional statements. That is, for nearly integrable nonlinear PDEs (under periodic boundary conditions) we do not know any result which is essentially infinite-dimensional. There are no doubts that such results exist. To find them is a big challenge. (10.1007/978-1-4020-6964-2_5)
  DOI : 10.1007/978-1-4020-6964-2_5
• Inégalités de Milnor-Wood géométriques
  
  • Besson Gérard
  • Courtois Gilles
  • Gallot Sylvestre
  Commentarii Mathematici Helvetici, European Mathematical Society, 2007, 82 (4), pp.753-803. We prove an extension of Milnor-Wood inequalities to a geometric situation. We study representations of the fundamental group of a compact manifold into the isometry group of a product of rank one spaces of the same dimension and show an upper bound on the volume of the representation. When the target group is the isometry group of the real hyperbolic space, we show the constance of the volume under deformations using the Schläfli formula and deduce a new and simple proof of a result of T. Soma; the result is that there are only finitely many closed hyperbolic three-manifolds dominated by a given closed three-manifold. (10.4171/CMH/109)
  DOI : 10.4171/CMH/109
• The nonlinear diffusion limit for generalized Carleman models: the initial-boundary value problem
  
  • Golse François
  • Salvarani Francesco
  Nonlinearity, IOP Publishing, 2007, 20 (4), pp.927-942. Consider the initial-boundary value problem for the 2-speed Carleman model of the Boltzmann equation of the kinetic theory of gases set in some bounded interval with boundary conditions prescribing the density of particles entering the interval. Under the usual parabolic scaling, a nonlinear diffusion limit is established for this problem. In fact, the techniques presented here allow treating generalizations of the Carleman system where the collision frequency is proportional to some power of the macroscopic density, with exponent in [-1,1]. (10.1088/0951-7715/20/4/007)
  DOI : 10.1088/0951-7715/20/4/007
• Extremal Kähler metrics on ruled manifolds and stability
  
  • Apostolov Vestislav
  • Calderbank David M. J.
  • Gauduchon Paul
  • Tønnesen-Friedman Christina W.
  Asterisque, Société Mathématique de France, 2008, 322, pp.93-150.
• Four lectures on KAM for the non-linear Schrödinger equation
  
  • Eliasson Hakan
  • Kuksin Sergei
  , 2008, pp.179-212. We discuss the KAM-theory for lower-dimensional tori for the non-linear Schrödinger equation with periodic boundary conditions and a convolution potential in dimension d. Central in this theory is the homological equation and a condition on the small divisors often known as the second Melnikov condition. The difficulties related to this condition are substantial when d≥ 2. We discuss this difficulty, and we show that a block decomposition and a Töplitz- Lipschitz-property, present for non-linear Schrödinger equation, permit to overcome this difficuly. A detailed proof is given in [EK06]. (10.1007/978-1-4020-6964-2_10)
  DOI : 10.1007/978-1-4020-6964-2_10
• New Asymptotic Profiles of Nonstationnary Solutions of the Navier-Stokes System
  
  • Brandolese Lorenzo
  • Vigneron Francois
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2007. We show that solutions $u(x,t)$ of the non-stationnary incompressible Navier--Stokes system in $\R^d$ ($d\geq2$) starting from mild decaying data $a$ behave as $|x|\to\infty$ as a potential field: u(x,t) = e^{t\Delta}a(x) + \gamma_d\nabla_x(\sum_{h,k} \frac{\delta_{h,k}|x|^2 - d x_h x_k}{d|x|^{d+2}} K_{h,k}(t))+\mathfrak{o}(\frac{1}{|x|^{d+1}}) where $\gamma_d$ is a constant and $K_{h,k}=\int_0^t(u_h| u_k)_{L^2}$ is the energy matrix of the flow. We deduce that, for well localized data, and for small $t$ and large enough $|x|$, c t |x|^{-(d+1)} \le |u(x,t)|\le c' t |x|^{-(d+1)}, where the lower bound holds on the complementary of a set of directions, of arbitrary small measure on $\mathbb{S}^{d-1}$. We also obtain new lower bounds for the large time decay of the weighted-$L^p$ norms, extending previous results of Schonbek, Miyakawa, Bae and Jin. (10.1016/j.matpur.2007.04.007)
  DOI : 10.1016/j.matpur.2007.04.007