Abstract-This technical note deals with a general class of discrete 2-D possibly nonlinear systems based on the Roesser model. We first motivate the introduction of Lyapunov type definitions of asymptotic and exponential stability. This will allow us to introduce and discuss several particularities that cannot be found in 1-D systems. Once this background has been carefully designed, we develop different Lyapunov theorems in order to check asymptotic and exponential stability of nonlinear 2-D systems. Finally we propose the first converse Lyapunov theorem in the case of exponential stability.

Abstract. We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are boundaries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kähler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived.

We give a topological interpretation of the space of L 2 -harmonic forms of finite-volume manifolds with sufficiently pinched negative curvature. We give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants. The method consists first in comparing L 2 −cohomology with weighted L 2 −cohomology thanks to previous works done by T. Ohsawa, and then in identifying these weighted spaces. RésuméNous donnons une interprétation topologique des espaces de formes harmoniques L 2 des variétés de volume fini,à courbure négative suffisamment pincée. Nous donnons des exemples montrant que cette interprétation n'est plus valable si la courbure n'est pas suffisamment pincée et que le résultat est optimal. La méthode utilisée consistè a comparer L 2 −cohomologie et L 2 −cohomologieà poids grâceà des travaux de T. Ohsawa, puisà identifier la L 2 −cohomologieà poids.

We show that on an open bounded smooth strongly pseudoconvex subset of C n , there exists a Kähler-Einstein metric with positive Einstein constant, such that the metric restricted to the Levi distribution of the boundary is conformal to the Levi form. To achieve this, we solve an associated complex Monge-Ampère equation with Dirichlet boundary condition. We also prove uniqueness under some more assumptions on the open set.

9 pages.International audienceIn this article, we prove a Lichnerowicz estimate for a compact convex domain of a Kähler manifold whose Ricci curvature satisfies $\Ric \ge k$ for some constant $k>0$. When equality is achieved, the boundary of the domain is totally geodesic and there exists a nontrivial holomorphic vector field. We show that a ball of sufficiently large radius in complex projective space provides an example of a strongly pseudoconvex domain which is not convex, and for which the Lichnerowicz estimate fails