Extremal metrics for the Paneitz Operator on closed four-Manifolds

TL;DR

This paper studies metrics on four-dimensional closed manifolds that optimize Paneitz operator eigenvalues, linking them to higher-order harmonic maps and extending surface case results to four-manifolds.

Contribution

It characterizes critical metrics of Paneitz eigenvalues as higher-order harmonic maps into spheres, extending known surface results to four-manifolds.

Findings

Critical metrics in conformal classes relate to extrinsic conformal-harmonic maps.

Extension of harmonic map characterization from surfaces to four-manifolds.

Partial characterization of general critical points of the eigenvalue functional.

Abstract

Let $(M^4,g)$ be a closed Riemannian manifold of dimension four. We investigate the properties of metrics which are critical points of the eigenvalues of the Paneitz operator when considered as functionals on the space of Riemannian metrics with fixed volume. We prove that critical metrics of the aforementioned functional restricted to conformal classes are associated with a higher-order analog of harmonic maps (known as extrinsic conformal-harmonic maps) into round spheres. This extends to four-manifolds well-known results on closed surfaces relating metrics maximizing laplacian eigenvalues in conformal classes with the existence of harmonic maps into spheres. The case of general critical points (not restricted to conformal classes) is also studied, and partial characterization of these is provided.