Four dimensional closed manifolds admit a weak harmonic Weyl metric

TL;DR

This paper introduces weak harmonic Weyl metrics on four-dimensional closed manifolds, proves their existence and uniqueness in conformal classes, and characterizes certain anti-self-dual metrics via quadratic functionals.

Contribution

It defines a new class of canonical metrics, proves their existence and uniqueness, and characterizes anti-self-dual metrics with positive Yamabe invariant.

Findings

Every closed four-manifold admits a weak harmonic Weyl metric.

The weak harmonic Weyl metric minimizes a specific quadratic functional in its conformal class.

Anti-self-dual metrics with positive Yamabe invariant are characterized by pinching conditions involving quadratic functionals.

Abstract

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.