By considering an abelian Chern-Simons model, we are led to study the existence of solutions of the Liouville equation with singularities on a flat torus. A non-existence and degree counting for solutions are obtained. The former result has an application in the Chern-Simons model.
Je présenterai les résultats d’une étude microlocale détaillée du spectre joint de deux opérateurs h-pseudo-différentiels qui commutent sur une variété de dimension deux en présence d’une singularité dite «focus-focus». L’étude couvre par exemple le cas du pendule sphérique étudié par Duistermaat, ou du fond de la bouteille de champagne, mais les phénomènes observés sont universels. On en observe principalement deux: une accumulation de valeurs propres au voisinage de la singularité en par rapport...
Nous introduisons de nouvelles régularités de Kuo-Verdier et montrons que pour une stratification
It is proved that the normal bundle of a distribution on a riemannian manifold admits a conformal curvature if and only if is a conformal foliation. Then is conformally flat if and only if vanishes. Also, the Pontrjagin classes of can be expressed in terms of .
We show that a certain eigenvalue minimization problem in two dimensions for the Laplace operator in conformal classes is equivalent to the composite membrane problem. We again establish such a link in higher dimensions for eigenvalue problems stemming from the critical GJMS operators. New free boundary problems of unstable type arise in higher dimensions linked to the critical GJMS operator. In dimension four, the critical GJMS operator is exactly the Paneitz operator.
It is proved that if an n-dimensional compact connected Riemannian manifold (M,g) with Ricci curvature Ric satisfying 0 < Ric ≤ (n-1)(2-nc/λ₁)c for a constant c admits a nonzero conformal gradient vector field, then it is isometric to Sⁿ(c), where λ₁ is the first nonzero eigenvalue of the Laplacian operator on M. Also, it is observed that existence of a nonzero conformal gradient vector field on an n-dimensional compact connected Einstein manifold forces it to...