Publications

2001

• Métriques kählériennes de volume fini, uniformisation des surfaces complexes réglées et équations de Seiberg-Witten
  
  • Rollin Yann
  , 2001. Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar curvature on M is equivalent to the parabolic polystability of E; moreover these metrics all come from finite volume quotients of $H^2 \times CP^1$. In order to prove the theorem, we must produce a solution of Seiberg-Witten equations for a singular metric g. We use orbifold compactifications $\overline M$ on which we approximate g by a sequence of smooth metrics; the desired solution for g is obtained as the limit of a sequence of Seiberg-Witten solutions for these smooth metrics.
• Fonctions zêta des hauteurs des espaces fibrés
  
  • Chambert-Loir Antoine
  • Tschinkel Yuri
  Progress in Mathematics, Springer, 2001, 199, pp.71-115. This paper is devoted to the estimation of the number of points of bounded height on fibrations in toric varieties over algebraic varieties, generalizing previous work by Strauch and the second author. Under reasonable hypotheses on ``Arakelov L-functions'' of the base, we show how to deduce a good estimate for the open subset of the total space of the unerlying fibration in torus. In passing, we improve drastically the error term for toric varieties themselves, generalizing a theorem by de la Breteche over any number field.
• Nearly Kähler 6-Manifolds with Reduced Holonomy
  
  • Belgun Florin
  • Moroianu Andrei
  Annals of Global Analysis and Geometry, Springer Verlag, 2001, 19, pp.307-319. We consider a complete six-dimensional nearly Kähler manifold together with the first canonical Hermitian connection. We show that if the holonomy of this connection is reducible, then the manifold endowed with a modified metric and almost complex structure is a Kählerian twistor space. This result was conjectured by Reyes-Carrión. (10.1023/A:1010799215310)
  DOI : 10.1023/A:1010799215310
• Vortex energy and vortex bending for a rotating Bose-Einstein condensate
  
  • Aftalion Amandine
  • Riviere Tristan
  Physical Review A : Atomic, molecular, and optical physics [1990-2015], American Physical Society, 2001, 64, pp.043611. For a Bose-Einstein condensate placed in a rotating trap, we give a simplified expression of the Gross-Pitaevskii energy in the Thomas Fermi regime, which only depends on the number and shape of the vortex lines. Then we check numerically that when there is one vortex line, our simplified expression leads to solutions with a bent vortex for a range of rotationnal velocities and trap parameters which are consistent with the experiments. (10.1103/PhysRevA.64.043611)
  DOI : 10.1103/PhysRevA.64.043611
• A splitting theorem for Kähler manifolds with constant eigenvalues of the Ricci tensor
  
  • Apostolov Vestislav
  • Draghici Tedi
  • Moroianu Andrei
  International Journal of Mathematics, World Scientific Publishing, 2001, 12, pp.769-789. It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler–Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kähler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kähler metrics of any even real dimension greater than 4. (10.1142/S0129167X01001052)
  DOI : 10.1142/S0129167X01001052
• From Kirchberg's inequality to the Goldberg conjecture
  
  • Moroianu Andrei
  , 2001, pp.283-292. The main result of this note is that a compact Kaehler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kaehler-Einstein manifolds. The problem of existence of Kaehler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the limiting case of Kirchberg's inequality for the first eigenvalue of the Dirac operator on compact Kaehler manifolds, as well as to the celebrated (still open) conjecture of Goldberg.