| Abstract: |
| Let $(M,g)$ be a closed Riemannian manifold of dimension $n \ge 3$ and $P_g$ be a conformally-covariant operator on $(M,g)$. We consider in this talk two problem at the crossroads of conformal geometry and spectral theory: 1) determining the extremal value that the renormalized eigenvalues of $P_g$ take as $g$ runs through a fixed conformal class and 2) determining whether these extremal values are attained at an extremal metric. Examples of such operators $P_g$ include the famous conformal Laplacian of the Yamabe problem, $P_g = \Delta_g + c_n S_g$, but also its higher-order generalisations of even order. Extremal metrics for these problems, when they exist, are not smooth in general, and yield interesting geometric objects, such as (singular) harmonic maps into large-dimensional spheres or least-energy possibly nodal solutions of prescribed $Q$-curvature type problems. Addressing these questions requires to investigate renormalized eigenvalues functionals that are highly non-smooth. We develop an ad hoc variational theory to study them: we obtain in particular semi-contiuity results and we provide an explicit Euler-Lagrange equation for local extremals. As a consequence we obtain new existence results for extremals by performing a bubble-tree analysis of suitable extremizing sequences. This is joint work with E. Humbert (Universite de Tours) and R. Petrides (Universite Paris Cite) and is based on arXiv:2505.08280. |
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