Mehdi Eddaoudi will speak in the geometry seminar on 02/11/2026. The talk will take place at 2pm in the Salle des Profs (9th floor of building NO). Mehdi’s title is “Critical metrics for eigenvalue ratios of geometric operators” and his abstract is below.
On a closed Riemannian manifold, it is well known that the problem of finding critical metrics for volume-normalized Laplace eigenvalues is closely connected to the existence of isometric immersions into spheres by eigenfunctions. In this talk, we discuss the same question for conformally covariant operators by considering ratios of eigenvalues instead. For the conformal Laplacian, we prove that critical points of the ratio of the first two eigenvalues correspond to spherical Einstein–harmonic maps. For the mixed Dirac–conformal Laplacian ratio of first eigenvalues, criticality leads to a spherical spinorial section satisfying an Einstein–Dirac-type equation. These results suggest a broader principle, namely that variational problems for ratios of eigenvalues of geometric operators naturally lead to field equations in mathematical physics.