On a closed Riemannian manifold, it is well known that critical metrics for volume-normalized Laplace eigenvalues are closely connected to minimal isometric immersions into spheres by eigenfunctions. In this talk, we address the analogous question for the conformal Laplacian by considering ratios of eigenvalues instead. We first motivate this normalization by establishing a Payne–Pólya–Weinberger inequality for the ratio of the first two eigenvalues in terms of a conformal invariant. Then we prove that critical metrics for this ratio correspond to harmonic–Einstein maps into spheres, which are solutions of a nonlinear sigma model coupled to gravity. Finally, we extend these results to the family of GJMS operators and beyond.

Dans ce travail en cours et en commun avec Ines Kath (Greifswald) et Georges Habib (Beyrouth), je m’attacherai à  décrire quelques propriétés des variétés riemanniennes portant une fonction satisfaisant une équation liant sa hessienne au tenseur de Ricci de la variété.

I will discuss some results extending classical (as well as more recent) theorems in algebraic geometry to (1,1) cohomology classes on compact Kahler manifolds. In particular I will discuss Nakamaye’s Theorem, the Fujita-Zariski Theorem, and Seshadri constants.