A sphere theorem on locally conformally flat even-dimensional manifolds

In this paper, we prove that a closed even-dimensional manifold which is locally conformally flat with positive scalar curvature, positive Euler characteristic and which satisfies some additional condition on its curvature is diffeomorphic to the sphere or projective space.     

On Perelman’s dilaton

By means of a Kaluza–Klein type argument we show that the Perelman’s F{\mathcal{F}} -functional is the Einstein–Hilbert action in a space with extra “phantom” dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his famous paper (Perelman in The entropy formula for the Ricci flow and its geometric applications, 2002). As a consequence the Ricci flow (modified by a diffeomorphism and a time-dependent factor) is the evolution of the “real” part of the metric under a constrained gradient flow of the Einstein–Hilbert gravitational action in higher dimension.     

Prescribing scalar and boundary mean curvature on the three dimensional half sphere

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half-three sphere, by deforming conformally its standard metric. Using blow-up analysis techniques and minimax arguments, we prove some existence and compactness results.     

A fourth order uniformization theorem on some four manifolds with large total Q-curvature

Given a four-dimensional manifold $(M,g)$, we study the existence of a conformal metric for which the Q-curvature, associated to a conformally invariant fourth-order operator (the Paneitz operator), is constant. Using a topological argument, we obtain a new result in cases which were still open. To cite this article: Z. Djadli, A. Malchiodi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).