Publications

2011

• Rigidity of Rank-One Factors of Compact Symmetric Spaces
  
  • Clarke Andrew
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2011, 61 (2), pp.491-509. We consider the decomposition of a compact-type symmetric space into a product of factors and show that the rank-one factors, when considered as totally geodesic submanifolds of the space, are isolated from inequivalent minimal submanifolds. (10.5802/aif.2621)
  DOI : 10.5802/aif.2621
• Extremal Kähler metrics on projective bundles over a curve
  
  • Apostolov Vestislav
  • Calderbank David M. J.
  • Gauduchon Paul
  • Tønnesen-Friedman Christina W.
  Advances in Mathematics, Elsevier, 2011, 227 (6), pp.2385-2424. Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$ admits a Kähler metric of constant scalar curvature if and only if $E$ is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of $E$ as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in sufficiently small Kähler classes. The methods used to prove the above results apply to a wider class of manifolds, called {\it rigid toric bundles over a semisimple base}, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds. (10.1016/j.aim.2011.05.006)
  DOI : 10.1016/j.aim.2011.05.006
• Extremal metrics on blowups
  
  • Pacard Frank
  • Arezzo Claudio
  • Singer Michael
  Duke Mathematical Journal, Duke University Press, 2011, 157 (1), pp.1-51. In this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of Kähler manifolds which already carry an extremal metric. As a particular case, we construct extremal metrics on $\mathbb {P}^2$ blown-up k points in general position, with $k \lt m+2$.
• Anticanonical divisors and curve classes on Fano manifolds
  
  • Hoering Andreas
  • Voisin Claire
  Pure and Applied Mathematics Quarterly, International Press, 2011.
• An application of coding theory to estimating Davenport constants
  
  • Plagne Alain
  • Schmid Wolfgang A.
  Designs, Codes and Cryptography, Springer Verlag, 2011, 61 (1), pp.105-118. We investigate a certain well-established generalization of the Davenport constant. For j a positive integer (the case j = 1, is the classical one) and a finite Abelian group (G,+,0), the invariant D_j(G) is defined as the smallest l such that each sequence over G of length at least l has j disjoint non-empty zero-sum subsequences. We investigate these quantities for elementary 2-groups of large rank (relative to j). Using tools from coding theory, we give fairly precise estimates for these quantities. We use our results to give improved bounds for the classical Davenport constant of certain groups. (10.1007/s10623-010-9441-5)
  DOI : 10.1007/s10623-010-9441-5
• Sums of dilates in groups of prime order
  
  • Plagne Alain
  Combinatorics, Probability and Computing, Cambridge University Press (CUP), 2011, 20 (6), pp.867-873. We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or -1 and if A < ℤ/pℤ is not too large (with respect to p), then | A + t A| is significantly larger than 2|A| (unless |t| = 3). In the important case |t| = 2, we obtain for instance | A+ tA| ≥ 2.08 |A|−2. (10.1017/S0963548311000447)
  DOI : 10.1017/S0963548311000447
• Endoscopy for real reductive groups
  
  • Renard David
  , 2011, pp.95-141.
• Toric geometry of convex quadrilaterals
  
  • Legendre Eveline
  Journal of Symplectic Geometry, International Press, 2011, 9, pp.343-385. We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kähler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.
• Entropy of quantum limits for symplectic linear maps of the multidimensional torus
  
  • Riviere Gabriel
  International Mathematics Research Notices, Oxford University Press (OUP), 2011, 2011 (11), pp.2396-2443. In the case of a linear symplectic map A of the 2d-torus, semiclassical measures are A-invariant probability measures associated to sequences of high energy quantum states. Our main result is an explicit lower bound on the entropy of any semiclassical measure of a given quantizable matrix A in Sp(2d,Z). In particular, our result implies that if A has an eigenvalue outside the unit circle, then a semiclassical measure cannot be carried by a closed orbit of A. (10.1093/imrn/rnq155)
  DOI : 10.1093/imrn/rnq155
• On an overdetermined elliptic problem
  
  • Pacard Frank
  • Hélein Frédéric
  • Hauswirth Laurent
  Pacific Journal of Mathematics, Mathematical Sciences Publishers, 2011, 250 (2), pp.319-334. A smooth flat Riemannian manifold is called an exceptional domain if it admits positive harmonic functions having vanishing Dirichlet boundary data and constant (nonzero) Neumann boundary data. In analogy with minimal surfaces, a representation formula is derived and applied to the classification of exceptional domains. Some interesting open problems are proposed along the way. (10.2140/pjm.2011.250.319)
  DOI : 10.2140/pjm.2011.250.319
• Renormalization and asymptotic expansion of Dirac's polarized vacuum
  
  • Gravejat Philippe
  • Lewin Mathieu
  • Séré Eric
  Communications in Mathematical Physics, Springer Verlag, 2011, 306 (1), pp.1-33. We perform rigorously the charge renormalization of the so-called reduced Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac operator, describes atoms and molecules while taking into account vacuum polarization effects. We consider the total physical density including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no 'real' electron. We show that it admits an asymptotic expansion to any order in powers of the physical coupling constant $\alphaph$, provided that the ultraviolet cut-off behaves as $\Lambda\sim e^{3\pi(1-Z_3)/2\alphaph}\gg1$. The renormalization parameter $0 (10.1007/s00220-011-1271-4)
  DOI : 10.1007/s00220-011-1271-4
• On the speed of approach to equilibrium for a collisionless gas
  
  • Aoki Kazuo
  • Golse François
  Kinet. Relat. Models, 2011, 4 (1), pp.87-107. We investigate the speed of approach to Maxwellian equilibrium for a collisionless gas enclosed in a vessel whose wall are kept at a uniform, constant temperature, assuming diffuse reflection of gas molecules on the vessel wall. We establish lower bounds for potential decay rates assuming uniform $L^p$ bounds on the initial distribution function. We also obtain a decay estimate in the spherically symmetric case. We discuss with particular care the influence of low-speed particles on thermalization by the wall. (10.3934/krm.2011.4.87)
  DOI : 10.3934/krm.2011.4.87
• The Cauchy problems for Einstein metrics and parallel spinors
  
  • Ammann Bernd
  • Moroianu Andrei
  • Moroianu Sergiu
  , 2011. We show that in the analytic category, given a Riemannian metric $g$ on a hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$ can be locally extended to a Riemannian Einstein metric on $Z$ with second fundamental form $W$, provided that $g$ and $W$ satisfy the constraints on $M$ imposed by the contracted Codazzi equations. We use this fact to study the Cauchy problem for metrics with parallel spinors in the real analytic category and give an affirmative answer to a question raised in Bär, Gauduchon, Moroianu (2005). We also answer negatively the corresponding questions in the smooth category.
• The Noether theorems. Invariance and conservation laws in the twentieth century
  
  • Kosmann-Schwarzbach Yvette
  , 2011, pp.205. (10.1007/978-0-387-87868-3)
  DOI : 10.1007/978-0-387-87868-3
• On the infinite fern of Galois representations of unitary type
  
  • Chenevier Gaëtan
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2011, 44 (6), pp.963-1019. Let E be a CM number field, F its maximal totally real subfield, c the generator of Gal(E/F), p an odd prime totally split in E, and S a finite set of places of E containing the places above p. Let r : G_{E,S} --> GL_3(F_p^bar) be a modular, absolutely irreducible, Galois representation of type U(3), i.e. such that r^* = r^c, and let X(r) be the rigid analytic generic fiber of its universal G_{E,S}-deformation of type U(3). We show that each irreducible component of the Zariski-closure of the modular points in X(r) has dimension at least 6[F:Q]. We study an analogue of the infinite fern of Gouvea-Mazur in this context and deal with the Hilbert modular case as well. As important steps, we prove that any first order deformation of a generic enough crystalline representation of Gal(Q_p^bar/Q_p) (of any dimension) is a linear combination of trianguline deformations, and that unitary eigenvarieties (of any rank) are etale over the weight space at the non-critical classical points. As another application, we obtain a general theorem about the image of the localization at p of the p-adic Adjoint' Selmer group of the p-adic Galois representations attached to any cuspidal, cohomological, automorphic representation Pi of GL_n(A_E) such that Pi^* = Pi^c (for any n).
• Almost complex structures on quaternion-Kähler manifolds and inner symmetric spaces
  
  • Gauduchon Paul
  • Moroianu Andrei
  • Semmelmann Uwe
  Inventiones Mathematicae, Springer Verlag, 2011, 184 (2), pp.389-403. We prove that compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians $Gr_2(C^{n+2})$. We also prove that irreducible inner symmetric spaces $M^{4n}$ of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces. (10.1007/s00222-010-0291-6)
  DOI : 10.1007/s00222-010-0291-6